Match ReportGylfi Sigurdsson scored a spectacular goal that proved decisive against his old club as the Blues eventually saw off the top flight's bottom club.
The Icelandic star ignited what was another largely dire performance with a wonderful curling strike in front of the Gwladys Street End that put Everton 2-1 up after Dominic Calvert-Lewin's equaliser in first-half stoppage time had cancelled out Leroy Fer's opener.
Wayne Rooney, whose spot-kick for the equaliser had come off the post via Lukas Fabianski's fingertips, added a third, again from the penalty spot, as Sam Allardyce oversaw his fourth win in five games and a return of 10 points from 12 in the Premier League.
The manager made no changes from the team that started at Newcastle, Sandro Ramirez making the bench with Oumar Niasse missing from the team sheet this time, but for 45 minutes his charges turned in a display horribly reminiscent of those under Ronald Koeman earlier in the campaign.
With two holding midfielders, they created hardly any chances and were back to conceding the first goal — from a set-piece no less — as well.
Despite losing Wilfred Bony to injury with just five minutes gone, Swansea had been rewarded for their industry and desire to get forward, particularly down the flanks when they took the lead 10 minutes before the break.
Everton had been forced to cede an uncomfortable amount of possession to the visitors in the early stages but and had just an attempted flick over Fabianski that sailed over and a routine Calvert-Lewin shot following impressive control on his chest from Holgate's long ball to show for their own efforts.
For the visitors, Narsingh had surged past Martina and cut inside but shot over the bar while a missed tackle by Schneiderlin in midfield led to Dyer sprinting through on the Blues' back line but he appeared to be foiled by a terrific sliding tackle from Holgate. Referee Jon Moss adjudged it be a foul, though, and booked the young defender to the disbelief and anger of the home crowd.
Pickford got down to save the resulting free kick but the sense of injustice galvanised the home faithful who had had little cheer up to that point.
Swansea took the lead, though, in frustratingly familiar circumstances. Tammy Abraham won a corner which was swung into the six-yard box where Fer arrived unmarked to convert.
Martin Olsson then hammered a long-range effort that flew inches past the post as Swansea looked to compound Everton's misery.
The first half was drifting to an unsatisfactory conclusion from Everton's point of view as it moved into first-half stoppage time but a throw to Rooney was moved on to Aaron Lennon, arguably the Toffees' man of the match, who jinked into the box and was tripped by Roque Mesa.
Rooney, who struggled to find his form all game, stepped up to take the resulting penalty and while his spot-kick was pushed onto the post by Fabianski, Calvert-Lewin followed up to ram the rebound home.
The second period was 11 minutes old before Everton had a chance to threaten the Swans' defence when Lennon won a corner but it came to nothing. Calvert-Lewin headed another corner well wide before Tom Davies came on for Schneiderlin with an hour gone and that helped change the pattern of the match from the hosts' perspective.
And Sigurdsson, having done a trial run earlier in the half from an almost identical location, scored the decisive goal not long afterwards. With Martina bombing past him as a decoy, the midfielder lined up a curling effort that he bent around Fabianski and inside the far post from 20-plus yards to bring Goodison to its feet.
Everton were still flattering to deceive as an attacking outfit but they were awarded another penalty when Jonjoe Kenny was tripped by Olsson as he drove past the fullback from Lennon's pass.
Replays suggested the offence took place right on the edge of the box but Rooney took the honours of the spot-kick again and this time he slammed home a more powerful effort than his first to make it 3-1.
Sigurdsson's neat pass between the two fullbacks with five minutes left put Calvert-Lewin in for one more opportunity but he was challenged as he shot and his effort was saved.
The football may not yet be pretty but Everton's improvement under Allardyce continues. The win marks a sixth game unbeaten for Everton in all competitions and moves them over Watford into ninth place, to within one point of Leicester and just six behind Tottenham Hotspur.
Everton return to Goodison Park following a brace of away games that yielded four points from six to play two home fixtures that present an opportunity to keep closing the gap on the top seven.
Three wins and a draw in the League since that dismal 4-1 reverse at Southampton have vaulted the Blues away from relegation danger and into the top half where they sit level on points with Watford and nine points off fourth place.
Now, with the team looking upwards rather than over their collective shoulder and first bottom club Swansea City then a fallible Chelsea team coming to Goodison Park before Christmas, manager Sam Allardcye is eyeing further improvement.
“The credit goes to the players who've achieved so much in a short period of time,” he said in his pre-match press conference. “10 points from 12 is a miraculous recovery from them.
“It's put the club in a very stable position that allows me to perhaps think a little bigger than when I first came in, to think more about sustaining our position in the top half, to think about improving the players because their confidence is greater, and taking each game as it comes and trying to get as many results as we can.
“The club's had a difficult spell and hopefully we can continue to get through this difficult spell. [The recent run of results] has given us some breathing space but not stopped us from being as ambitious as we possibly can.”
In view of Everton's improved position in the table and the feeling that the club is getting close to being back on track in terms of its European aspirations, the focus will increasingly be on how the team plays under Allardyce given that he has implemented a very back-to-basics approach since taking charge.
That has firmed up what was one of the softest defences in the Premier League but has come at the expense of genuine entertainment going forward, particularly away from home in the Merseyside derby and at Newcastle in midweek.
It may be too early for radical change in that respect but there was a noticeable attempt to keep the ball against the Magpies in the Blues' 1-0 win which was aided by what was arguably Morgan Schneiderlin's best performance of the season.
The Frenchman has been curiously and frustratingly poor this season and lost his place in the side amid a dreadful run of results for Everton and suggestions that he lacked focus and discipline. Since coming on at half time against Liverpool last Sunday, the £20m acquisition from Manchester United has looked more like the composed central midfielder he was when he first arrived almost a year ago.
That will likely see him keep his place alongside Idrissa Gueye for the time being, despite many supporters' misgivings about playing two holding midfielders, particularly at home.
Assuming he plays, Schneiderlin's selection would likely be at the expense of Tom Davies while Wayne Rooney continues in his more withdrawn role playing off the improving Gylfi Sigurdsson and behind Dominic Calvert-Lewin. The young striker has earned his stripes leading the line and will probably be preferred to Oumar Niasse again should Allardyce only opt for one striker.
Swansea, of course, come to Goodison on a poor run of form that has seen them sink to the bottom of the table. Just one away win all season so far and one victory since mid-October has brought the Welsh club's top-flight future into sharp focus and highlighted just how important Sigurdsson and Fernando Llorente were in keeping them up last season.
Despite that, there are no easy games in the Premier League, a fact that Allardyce knows all too well and he has been cautioning his players against assuming the points are theirs for the taking.
“I'm not expecting any complacency,” he said. “That's been my big word to the players — that [Swansea] this will not be an easy task [so] don't slip up in against a side that everybody expects you to beat.
“You'll only get the opportunity to beat them if you play 100% to your level which is what we've had to do over last the four games we've played.”
Kick-off: 8pm, Monday, 18 December, 2017
Referee: Jonathan Moss
Last Time: Everton 1 - 1 Swansea City
Predicted Line-up: Pickford, Kenny, Holgate, Williams, Martina, Gueye, Schneiderlin, Rooney, Lennon, Sigurdsson, Calvert-Lewin