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The £45m midfielder received a Wayne Rooney pass and cut in from the left-wing before letting fly with a magnificent - and completely unstoppable - 64th minute shot.
That brought Everton their first lead of the night after bright young England starlet Dominic Calvert-Lewin had levelled Leroy Fer’s shock opener.
The worst attack in the Premier League looked even worse after just three minutes when Wilfried Bony hobbled off with a pulled muscle.
But that did not prevent Paul Clement’s side from taking a 35th minute lead after a moment of disastrous defending from the Toffees.
Tom Carroll’s right-wing corner was met by the unmarked Fer who sprinted in and converted from just inside the six-yard box.
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Thankfully, for Everton at least, there is a reason Swansea had come to Goodison on the back of five straight away defeats and they showed it in first half injury-time.
Roque Mesa made a clumsy attempt to tackle Aaron Lennon in his area and succeeded only in tripping the winger.
Rooney took the resulting pen and watched in horror as Lukasz Fabianski touched it onto his post with a superb diving save.
But youngster Dominic Calvert-Lewin was alert and skilful as he latched onto the rebound and drove an unstoppable shot into the net.
When home fans had seen Bony - the only Swan to have scored in their last seven games - limp off, they must have been banking on a clean sheet.
But there were early warning signs, with a couple of long-range blasts from Luciano Narsingh suggesting the Welsh side might be in the mood to add to their paltry nine goals so far this season.
And, after Fer had made it 10, full-back Martin Olsson did not miss by much with a terrific strike from well outside the area.
Yet the bigger concern for Allardyce, looking to add to a run of three wins and a draw since he officially took over from Ronald Koeman, was the lack of ideas at the other end.
Rooney - whose nine goals before last night matched Swansea’s entire output - looked the best hope of sparking Everton into life.
It took until the 19th minute for the hosts to carve out anything resembling a chance, a Rooney lob high over the goal from Cuco Martina’s cross.
Calvert-Lewin followed that up by threading a shot through a crowded area from 20 yards with Lukasz Fabianski smothering it easily.
Lennon finally brought the crowd to life, just before the half-hour, when he received a short Mason Holgate pass and shoved the ball through the legs of Alfie Mawson before running on and planting a shot just wide.
And Holgate was involved again soon after, going into the book after upending Nathan Dyer and presenting Swansea with a free-kick from which Carroll beat the wall but not Pickford.
The second half started as tamely as the first although Calvert-Lewin sparked the crowd into life when he charged down a Fabianski clearance.
And Sigurdsson tried his luck against his former employers with a 25-yard shot that the keeper smothered easily before the lively Calvert-Lewin headed wide.