Sigurdsson left Lukasz Fabianski with no chance in the second half by bending a wonderful effort beyond the Pole from outside the box to give Everton a 2-1 lead at Goodison Park.
The Toffees killed off the clash later in the half as Wayne Rooney dusted himself down after missing a first-half penalty to score from the spot on 74 minutes.
But Allardyce was left purring over what the Iceland international did prior to that.
He told ESPN: "It's just what you need. Changes the game. You need a bit of genius. You need a bit of work from a player in the squad who can change a game and that's what we got.
Leroy Fer had fired Swansea into a surprise 1-0 lead, but Dominic Calvert-Lewin reacted quickest to Fabianki's penalty save from Rooney to slide home an equaliser.
Consequence of the win, the Merseysiders are now ninth in the Premier League, having been one point above the relegation zone just a few short weeks ago.
And if Everton can emerge from the festive period unscathed then who knows, maybe Allardyce can help the Toffees salvage something from what looked certain to be a disastrous season under Ronald Koeman earlier in the campaign.
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