Everton eased the pressure on their shoulders from their faltering league campaign with a solid victory at Hillsborough that booked their place in the fourth round of the Carabao Cup.
Dominic Calvert-Lewin struck twice inside the first 10 minutes to put the Blues into a lead they held to the final whistle despite a couple of scares from dead-ball situations from which Sheffield Wednesday will feel they should have profited.
The result will come as huge relief to manager Marco Silva whose position has been under scrutiny once more in the wake of three defeats in four Premier League games.
The Portuguese made five changes to the line-up, bringing in Calvert-Lewin, Alex Iwobi, Tom Davies, Mason Holgate and Djibril Sidibé for Moise Kean, Gylfi Sigurdsson, Morgan Schneiderlin, Michael Keane and Seamus Coleman.
Having conceded in the first minute at Lincoln in the previous round, Everton almost shot themselves foot early again when Davies gave the ball away in a dangerous area but he was bailed out by his goalkeeper, Jordan Pickford making a vital save from Sam Winnall.
The Blues were ahead a few minutes later, though. Sidibé galloped into space down the right and he picked out Calvert-Lewin with a brilliant cross where the young striker controlled and rifled home in off the underside of the crossbar.
It was 2-0 in short order with just 10 minutes on the clock. Bernard and Iwobi combined and the Nigerian's centre off the Brazilian's lovely disguised pass found Calvert-Lewin who slid in to steer it home.
Lucas Digne, skipper for the night, then won a corner on the left but Richarlison put his header wide and the Frenchman almost set Calvert-Lewin up for his hat-trick but his free-kick delivery was just too far ahead of him.
The Toffees' vulnerabilities from corners resurfaced midway through the first-half after a mix-up between Mina and Pickford but Dominic Iorfa missed with a free header that should have halved the home side's deficit.
Then Pickford had to save from Adam Reach and Luongo drove a shot narrowly wide as Wednesday started to get a foothold in the game.
Calvert-Lewin had his third on a plate after Delph fed Richarlison following a driving run but he headed too close to Cameron Dawson who pushed his header behind well and the striker almost had another chance from the resulting corner but Sheffield escaped.
Everton made a quiet start to the second half and when Delph gave the ball away cheaply 10 minutes in, Winnall was in but chose the wrong option and passed up a gilt-edged chance for the home team.
Another set-piece and another free header was squandered by Winnall from the edge of the six-yard box with 62 minutes on the clock as Everton's weak dead-ball defending threatened to be their undoing once more.
Silva took Richarlison off with 23 minutes left and introduced Theo Walcott but it was Sidibé who tried to trouble Dawson but he couldn't keep his effort on target.
Bernard then made way for Sigurdsson with a quarter of an hour to go and the Icelandic international quickly seized on a bad pass but Calvert-Lewin couldn't capitalise on his ball into the box.
At the other end, Reach unloaded an impressive shot that whistled past Pickford's goal as Wednesday tried to find a way back into the game with five minutes left and Digne went close at the other end but his shot deflected behind.
Though new Wednesday boss Garry Monk made a slew of changes to his team, Everton could only have beaten what was in front of them and in that respect it was job done despite a largely uninspiring second-half display.
The match will have been especially sweet for Calvert-Lewin who has now started three League Cup ties for the Toffees and grabbed a brace in each of them.
Everton travel to Hillsborough for a Carabao Cup third round tie that has taken on huge significance for manager Marco Silva following Saturday's shock home defeat to Sheffield United.
The Blues make the trip to Sheffield Wednesday for what will be the first time they have faced the “steel city” clubs in successive games for 51 years and the Portuguese will be looking for a positive reaction from his players, lest another early cup exit heighten the scrutiny of his tenure.
Everton are reeling from a run of three defeats in four games in the Premier League but have an opportunity to steady the ship and bolster morale if they can navigate their way into the last 16 of the League Cup.
Having made somewhat heavy work of despatching League One Lincoln City in the last round, the Blues take on Championship opposition this time around. Wednesday, who beat Rotherham away in the second round, come into the tie sitting in ninth place in their division with four wins from their first eight league games.
It will mark the third match in charge for new boss Garry Monk, the former Swansea City and Birmingham boss who replaced Steve Bruce as manager of the South Yorkshire side earlier this month.
Monk predictably painted Everton, as the top-flight outfit as the favourites to win the tie, describing it as a “free hit” for his players against more fancied opposition. Silva, meanwhile, has called on his charges to be more aggressive and to show 100% commitment after losing meekly to Bournemouth and Sheffield United in the past 10 days.
All eyes will be on the Portuguese's team selection as he seeks a route out of his team's current malaise. He made a handful of changes for the trip to Lincoln and may feel moved to again, with the likes of Tom Davies, Djibril Sidibé, Dominic Calvert-Lewin, Mason Holgate and Alex Iwobi itching to make starts.
Iwobi, in particular, is expected to be in the XI after being named on the bench on Saturday while Sidibé was handed his debut in the previous round and could make only his second appearance since arriving on loan tomorrow evening. Whether Iwobi comes in in place of Bernard or the under-performing Gylfi Sigurdsson remains to be seen but the Iceland international was able to use the match in the last round to find some form, albeit temporarily.
As far as Davies is concerned, there has been a clamour for the young midfielder to be given a chance to impress in a part of the pitch that has become a concern this season and where André Gomes is expected to be missing again because of a rib injury. Davies has made just one fleeting appearance as a substitute so far and surprisingly absent from the matchday squad in the last round. With Silva searching for solutions in central midfield, the England U21 man could get his chance at Hillsborough.
Up front, the manager has a decision to make over whether to persist with Moise Kean, a player unfairly scapegoated in some quarters for Everton's lack of goal threat on Saturday, to recall Calvert-Lewin or go with another option altogether like moving Richarlison back up top, a tactic that was used with mixed results last season but which could be effective against lower-division opposition.
Regardless of which team he puts out, Silva knows that his side should be good enough to beat a Championship side, even away from home and he will be afforded little slack if his team aren't able to get the job done. With Champions, Manchester City, coming to Goodison Park on Saturday and a tricky visit to Turf Moor looming the following weekend, Everton need to shore things up at the back, find their mojo going forward and start grinding out results again.
Kick-off: 7:45pm, Tuesday, 24 September, 2019
Referee: Jeremy Simpson
Last Time: Sheffield Wednesday 0 - 3 Everton (September 2007)
Predicted Line-up: Pickford, Sidibé, Keane, Mina, Digne, Delph, Davies, Sigurdsson, Iwobi, Richarlison, Calvert-Lewin
Note: There is no televised coverage of this game in any broadcast market