Everton 3 - 1 Swansea City
Gylfi Sigurdsson's trademark strike made it 2-1 against his old club and Everton eventually prevailed comfortably
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.