Everton's increasingly worrying lack of cutting edge threatened to expose them to the lottery of penalty kicks as battling Stevenage almost took them to the wire before Marouane Fellaini came off the bench to edge the Blues through to the third round.
A team featuring no fewer than eight changes from the one that had ground their way to a goalless stalemate against West Bromwich Albion on Saturday dominated their League One opposition but, in what has become a feature of Roberto Martinez's reign so far, they could not find the killer instinct in the final third to press home that superiority.
In truth, they should have been comfortably into the next round well before the end of the regulation 90 minutes and surely would have been were it not for dogged defending by tge visitors, some questionable decision-making by Everton in and around the penalty area, and some wasteful finishing by Arouna Kone.
The Ivorian made his first start since joining the Blues as Martinez's biggest signing of the summer thus far but he looked for all the world like was still in pre-season mode. An 11th-minute opportunity less than a yard from goal that somehow stayed out set the tone for his evening and though he would get the entire 120 minutes to stake his claim to a regular starting role, he couldn't convert the goal to spare Everton's blushes.
That honour fell to the man who, if Manchester United and the press get their way, might not even be an Everton player this time next week and he underscored how much he would be missed by being in the right place at the right time to steer home the winner.
The Blues had already looked more potent when he replaced Ross Barkley six minutes into injury time, the youngster's early string-pulling influence having waned considerably in the second half of normal time, and after he narrowly missed with a bullet header from a corner in the 112th minute, he would make no mistake when the ball bounced fortuitously to him off Steven Naismith eight yards from the Stevenage goal.
It was tough luck on Stevenage who had worked tirelessly to close Everton down and keep them at arm's length for much of the contest. As they has been on Saturday, the Blues were full of neat passes and probing balls around the penalty area but the final ball was again lacking quality.
The brightest moments of a first half that seemed to be going to plan until Luke Freeman crashed home the opening goal in the 36th minute came from Everton's two 19 year-olds. Barkley was everywhere for most of the 45 minutes, water-carrying from defence to attack one minute, unleashing a stinging drive from 20 yards the next as he did when he combined with Deulofeu just outside the area in the seventh minute.
The Spaniard, meanwhile, was keen to display his box of tricks down the right flank. Initially it mostly consisted of knocking the ball past the fullback Deacon and skinning him for pace before crossing from the byline which he did to great effect to serve up that first chance for Kone. The striker could not stab home the ball as it bobbled around by the back post, though, and Stevenage escaped.
10 minutes later, another piece of magic by Deulofeu further in-field almost produced the opener when he took one touch wide of his mark and bent a lovely shot that narrowly missed the far post, before another low cross from the right after Deacon had been left for dead again went begging.
And the Barcelona teenager's efforts were for nought just past the half hour when Chris Day flapped at his cross and the ball fell invitingly in front of Leon Osman. The veteran midfielder, who should have controlled this encounter against a lower-division side but who, instead, served up more fuel for those detractors who feel his best days are well behind him, made a mess of the chance, though, with an awkward mis-kick with the goal gaping.
Stevenage hadn't threatened Joel Robles much in the Blues' goal but the home side did escape in the 13th minute when Dougherty's shot struck a defender before it could trouble the Spanish 'keeper. That proved to be a warning for Everton, though, because when Tounkara was allowed to chase down a long ball forward unimpeded and cut the ball back across the box, Day knocked the ball on towards Freeman who took one touch before hammering into the roof of the net to make it 1-0.
Everton resumed their patient control of proceedings and probably should have leveled in the 45th minute when Kone was set up nicely by Osman but he ballooned his shot wildly over the bar.
Thankfully, Deulofeu's precocious self-confidence kicked in in first-half stoppage time and, taking one more run at the Stevenage defence as they backed off, he turned his man around the 18 yard line and stroked an inch-perfect shot in off the inside of the post.
The second half promised much in the early going but eventually offered up nothing but more frustration for an increasingly uneasy home crowd. Barkley forced a good save from Day who pushed away his excellent shot at full stretch and a wonderful cross by Deulofeu almost put Naismith in at the back post but, on the stretch, the Scot couldn't get decent contact on the ball to turn it home.
Then, Naismith profited from a fortunate bounce of the ball and surged forward, slipping it into the path of Kone who tried to clip a shot past the 'keeper but Day foiled him to deny ano almost certain goal. If the Ivorian had been sloppy with his finishing before, this was much better and he was unlucky not to notch his first goal in Everton colours.
And the visitors' goal appeared to be living a charmed life as first Barkley's low ball flew across the face of goal with Kone a yard too far behind to try and slide it home, and then a chance opened up for Osman in front of goal but his strike cannoned off a defender and bobbled behind.
But that early Everton momentum started to ebb away as the half progressed, with some of the distribution of the ball becoming increasingly erratic. Barkley's influence had more or less evaporated by the midway point and Deulofeu was increasingly becoming a marked man, both in terms of numerical attention focused on him and the kinds of cynical challenges on him aimed at throwing him out of his rhythm.
Bryan Oviedo, who played the full game at left back, betrayed some inexperience and profligacy in possession and showed that he isn't yet ready to take over from Leighton Baines and even Seamus Coleman, who came on for Tony Hibbert after 65 minutes, had limited impact on the other flank. John Stones was hugely impressive as part of a back three, though, alongside John Heitinga and, first, Phil Jagielka and then Sylvain Distin.
An unwanted 30 minutes of extra time beckoned, though, as Deulofeu shot straight at Day with five minutes left and was then denied in full flight when Gray got a toe to the ball to force a corner, the defender benefiting from sharp eyes by referee Keith Shroud who waved away claims for a penalty.
It would take an intervention from Robles, though, to prevent Stevenage grabbing a dramatic winner with almost the last kick of the 90 minutes. Greg Tansey let rip wit a dipping shot from 20 yards that might have crept under the bar but the 'keeper tipped it over the bar.
To be frank, most of the extra time period was desperately poor and sluggish fare from Everton who retained parity in the 93rd minute when Freeman waltzed through unchecked again but his shot was stopped by Distin.
Heitinga bobbled a shot disappointingly wide at the other end a few minutes later before Deulofeu came to life again with another bursting run down the right to latch onto Osman's pass down the touchline. The Spaniard's low cross caused panic in the home defence as Gray slid crotch-first into the near post but although Kone was eventually able to fashion a chance from the loose ball, he scuffed his boot over the top of the ball and it was hacked clear.
Either the number 9 or the equally up-and-down Naismith could have been withdrawn for Fellaini as the third and final substitution but it was Barkley who had given way and the Belgian made the difference. First he forced a corner with a low shot that the sprawling Day pushed behind, then he despatched that header just wide before he pounced four minutes later to send Everton through on a wave of relief.
Few will have been more relieved than Martinez who finally notched his first competitive win as Everton manager. Though none of the players he started were huge gambles, the number of changes arguably was but the eleven he sent out had enough of the ball and fashioned enough half-chances to have won the game comfortably at the first time of asking.
That they didn't, that they again struggled to score the goals they needed, will keep the boss busy in the coming weeks as he implements the changes he wants to make to the squad and approach he inherited from David Moyes. What matters is that Everton are in the hat for the third round but with the transfer window still open, the evidence of this close call suggests that Martinez might still need to add some quality to a team lacking a real cutting edge.
Roberto Martinez steps onto the road to Wembley for the first time as Everton manager as the Blues take on Stevenage at Goodison Park in the Capital One Cup second round.
The Spaniard becomes the latest Toffees boss to attempt to win the one major domestic trophy that Everton have yet to win; his predecessor, David Moyes, got as far as the semii-final stage in 2008 but, as would be the case in the FA Cup Final the following year, he couldn't push his team past Chelsea.
It was a frustrating loss at Leeds last season that stained Moyes's last season with the Blues, a game in which he shot himself in the foot with a raft of changes from his regular League line-ups.
Perhaps emboldened by an emphatic win over Leyton Orient in the second round at Goodison, Moyes introduced of a number of young players from the start of the match who couldn't cope with a spirited and energetic performance by the home side. The result was 2-1 defeat and a premature exit from the problem.
With the luxury of a home tie, Martinez is expected to ring some changes as well, but probably not to the same extent. Gerard Deulofeu is in line for his first competitive start after watching the first two games of the season from the bench and his manager admitted that the Spaniard has been frustrated at being left out to this point.
An outing against lower-division opposition provides the Spanish teenager a safer introduction to the English game and, after an unrevealing performance in the heat of Miami against Valencia in the International Champions Cup earlier this month, Evertonians will be looking forward to getting their first proper look at him.
Another player expected to make his first start of the campaign is Victor Anichebe, who could get the nod ahead of Arouna Kone, while John Stones might also get some game time after impressing so much on the pre-season tour of the United States and John Heitinga could figure, too.
Stevenage have made a chequered start to the new season, losing two and winning two of their first four League One fixtures but it was a home victory over Championship side Ipswich Town that earned them passage to the second round of the League Cup.
Sensibly, Martinez says he won't be underestimating the visitors as he looks for his first victory since assuming the Everton hotseat.
|Premier League Scores|
|Premier League Table|
|2013-14 Reports Index|
|Hibbert (66' Coleman)|
|Jagielka (46' Distin)|
|Barkley (96' Fellaini)|
|Subs not used|
|Smith (53' Tansey)|
|Doughty (89' Heslop)|
|Charles (27' Tounkara)|
|Subs not used|