The 28-year-old Icelandic international left Swansea for a record £45 million ($60 million) in the close season and his 30-yard effort in the second half illustrated why Everton wanted him so badly.
Dominic Calvert-Lewin, with his seventh goal in all competitions, had brought Everton level after Leroy Fer had given the visitors the lead with his first goal in 38 matches.
Wayne Rooney wrapped up the points with a penalty — he had an earlier one saved before Calvert-Lewin knocked in the rebound — for his 10th goal of the season.
Victory sees Everton rise to ninth — a heady spot given they were in the relegation places on October 23 — and unbeaten in five matches under Allardyce.
“In games you need something special and Gylfi provided that,” Allardyce told Sky Sports.
“From there it was nice and comfortable.
“It is a great three points considering how poorly we played in the first half.”
Swansea, who lost their sixth successive away match and have only one win in their last 10 games, are four points adrift of safety.
“The mood in the dressing room is very flat,” Swansea manager Paul Clement told the BBC.
“We’re bottom of the table on 12 points. We’ve got to focus on getting to 15 points at the halfway mark.”
– ‘I’m not giving them up’ –
The visitors suffered a huge blow when Ivory Coast striker Wilfried Bony pulled up injured with what Clement believes is a hamstring problem and had to go off in the fifth minute.
It was the visitors, though, who broke the deadlock.
Jordan Pickford could do nothing about Fer’s close range half-volley after the Dutchman had escaped Ashley Williams’s attention, nipping around the back of the defender to give the visitors the lead with only their 10th goal of the campaign.
However, the sides went in at half-time all square as in time added on Roque Mesa felled Aaron Lennon to concede a penalty.
Lukasz Fabianski did superbly to save Rooney’s spot-kick but Calvert-Lewin followed up to rifle the ball home.
Rooney, though, said he is not going to surrender penalty duties even though he has the second worst miss record in Premier League history — 10 from 33 with England and Newcastle great Alan Shearer missing 11 of 67.
“I’m a confident person and believe in myself,” Rooney told Sky Sports.
“If you look at the pens I have missed you will see I put a bit of top spin on the ball so someone could follow up! I’m not giving them up.”
There was not a lot to choose between the sides for the first 20 minutes of the second period until Sigurdsson came up with his moment of magic.
He came in from the left wing after receiving the ball from Rooney and unleashed an unstoppable right-footed curling effort into the far top corner.
Swansea’s luck was well and truly out when Jonjoe Kenny went down in the area under a challenge by Martin Olsson 18 minutes from time and referee Jon Moss pointed to the spot despite the contact being outside the box.
Rooney this time made no mistake, though Fabianski got a hand to his powerfully-hit spot-kick.
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