Emmanuel Adebayor broke the deadlock after just 34 seconds for Spurs but Everton quickly levelled thanks to Phil Jagielka's header.
Kevin Mirallas made it 2-1 to the visitors with a wonderful solo effort for the second time in as many weeks, but Sigurdsson snatched a point for Tottenham three minutes from time.
Tottenham may have been without talisman Gareth Bale due to an ankle injury, but they made an ideal start as Adebayor tucked away a delightful left-wing cross from Jan Vertonghen inside the opening minute after Sylvain Distin had missed the interception.
Everton responded strongly and equalised in the 15th minute when Jagielka climbed above Vertonghen and powered a header through the legs of goalkeeper Hugo Lloris.
Spurs finished the first half on top and felt they should have had a penalty when Clint Dempsey's cross crashed into Darron Gibson, who had his arms raised above his head, but the officials waved play on and replays suggested the ball had hit his chest or shoulder.
Everton went in front in the 53d minute when Mirallas, who scored a wonderful winner against Stoke last weekend, worked his magic again, weaving one way and then the other past a couple of defenders before firing a terrific low finish through the legs of Steven Caulker and past Lloris.
There was almost a leveller for Tottenham moments later as Mousa Dembele's shot from 20 yards took a deflection which could have easily wrong-footed Tim Howard, but the keeper managed to get something on it and parry the ball onto the crossbar.
Spurs kept pressing but were becoming frustrated by Everton's resilient defending and were restricted to long-range strikes from distance as Adebayor, Dempsey and Kyle Walker all tried their luck.
Eventually the hosts were rewarded in the 87th minute as Walker darted down the right wing and cut the ball back for Adebayor, whose shot hit the post but fell kindly for Sigurdsson to score.
The result keeps Spurs six points clear of sixth-placed Everton, having played a game more, as the race for UEFA Champions League qualification goes down to the wire.