The Iceland international, who left the Swans in a £45m summer move, scored the vital second after Dominic Calvert-Lewin had cancelled out Leroy Fer’s opener by converting Wayne Rooney’s saved penalty on the stroke of half-time.
Rooney then made sure of the points with his sixth goal in five games when the hosts were awarded a fortuitous second spot-kick in the 72nd minute.
It left Clement, who took over on January 3 with the club bottom of the table and guided them to safety, battling to save his job as shot-shy Swansea currently have a similar record to the one which led to Bob Bradley’s sacking almost a year ago.
Just 12 points from 18 matches, and more worrying just 10 goals scored, leave the Welsh club four points from safety.
The opposite is true for Everton, who are upwardly mobile under Sam Allardyce with four wins and a draw from his first five matches in charge, earning 10 out of 12 points to lift his side from relegation contenders to ninth place in less than three weeks.
A dreary opening half-hour, punctuated only by injury to Wilfried Bony less than four minutes in, was belatedly ended by Aaron Lennon screwing a shot wide after nutmegging Alfie Mawson.
Goodison awoke when referee Jon Moss booked Mason Holgate for a foul on Nathan Dyer despite there appearing to be no contact and only Jordan Pickford saving Tom Carroll’s free-kick prevented more fury.
However, there was plenty of that aimed at former Swans defender Ashley Williams five minutes later when he lost Fer at a corner and the unmarked midfielder stabbed home Carroll’s inswinging corner from four yards.
It was Fer’s first Premier League goal in 13 months and Swansea’s first not scored by Bony since October 28.
Martin Olsson’s low drive flew just wide as the visitors looked to capitalise on their advantage but it was soon wiped out when Roque Mesa tangled with Lennon and Moss pointed to the spot.
Rooney’s effort was tipped onto the post by Lukasz Fabianski — the 10th time in 32 attempts England’s record goalscorer has failed from the spot in the Premier League — but Calvert-Lewin converted the rebound.
The pivotal goal, when it came in the 64th minute, could not have been a surprise for Swansea as former favourite Sigurdsson, collecting a pass from Rooney wide on the left, cut inside to brilliantly curl a shot inside the far post from 25 yards.
When you are down on your luck there is no respite and Rooney blasted home his 10th of the season after Moss ruled Olsson’s foul on Jonjoe Kenny was inside the area despite contact being made well outside.
Pickford, Kenny, Holgate, Williams,Martina, Gueye, Schneiderlin (Davies 61),Lennon (Lookman 79), Rooney (Sandro 89), Sigurdsson, Calvert-Lewin.
Fabianski, Naughton, Fernandez, Mawson, Olsson, Fer, Roque , Carroll (Clucas 81), Narsingh,Bony (Abraham 5),Dyer (Ayew 74).