The Toffees looked like taking all three points from north London after Phil Jagielka and Kevin Mirallas scored following Emmanuel Adebayor's 35th second opener.
But Sigurdsson, charged with the task of playing in the hole where Gareth Bale, who missed the game through injury, has been so effective this season, salvaged a point for the hosts by scoring a crucial goal three minutes from time.
Despite finding the net twice, Tottenham sorely missed the scoring talents of Bale and the creative wizardry of Aaron Lennon.
In the two days that have passed since Bale was carried off on a stretcher against Basle, Andre Villas-Boas has insisted that he had enough strength in depth to survive without the services of the Welshman, and winger Lennon.
That was not the case on Sunday, however. Tottenham lacked creativity going forward and often had to resort to long shots. Although he performed with the energy he has often lacked this season, Adebayor spent much of the 90 minutes in Sylvain Distin's back pocket.
Spurs rarely looked like breaching the Toffees back four, but Sigurdsson's last-gasp equaliser led to a sense of relief at the final whistle from the Tottenham fans.
The goal will not mask Tottenham's precarious predicament, though.
Put simply, if Arsenal beat Everton in their game in hand, they will leapfrog their bitter rivals in the hunt for Champions League qualification.
And with games against Chelsea and Manchester City to come in their final six games, there is an inescapable feeling that Tottenham are in the process of throwing away their Champions League dreams once more.
All the pre-match talk had been about how Spurs would struggle without Bale's goals, but they gave the perfect response by breaking the deadlock with the second quickest goal of the season.
In the first move of the match, Everton's defence backed off Jan Vertonghen, he delivered a teasing ball across the box, and Adebayor stuck his right leg and prodded the ball past Tim Howard.
Everton were shell-shocked by the goal, but it did not take them long to come back in to the game.
Michael Dawson almost gifted them a 10th-minute equaliser when he passed the ball straight to Leon Osman 15 yards out, but he curled high and wide.
Dawson then cut out Victor Anichebe's cross as a row of Everton attackers lined up to convert, but unfortunately for the Tottenham captain, his defensive colleague Vertonghen let him down from the resulting corner.
The Belgian lost Jagielka at the back post and he nodded Leighton Baines' cross in to the bottom corner.
Tottenham's temper threatened to boil over. Kyle Walker, dismayed at a poor pass from Mousa Dembele, shouted at the Belgian furiously and also gestured angrily at Lewis Holtby.
Villas-Boas urged his players to regain their composure and they did. Howard remained untested though, as long-range efforts from Dembele, Dempsey and Holtby all flew well wide.
Spurs thought they should have had a penalty on the half four when Dempsey's cross brushed Darron Gibson's arm, but Andre Marriner waved play on.
Everton sounded an early warning just after the restart when Darron Gibson struck a curling shot just wide, and with 52 minutes on the clock, they were ahead.
Mirallas drifted in from the right flank and drifted past Scott Parker and Steven Caulker before slotting the ball beyond Hugo Lloris.
Spurs went straight on the attack and almost scored an equaliser through Dembele, whose 20-yard shot ricocheted off John Heitinga's foot and Howard's arm on to the bar.
Heitinga almost sealed the win at the other end soon after but Dawson put in a crunching goal-saving tackle to deny the Dutchman.
Howard punched a 25-yard effort from Caulker away on the hour as Spurs continued to resort to long shots on goal. Dempsey went down in the box and earned a booking for diving.
The level of Tottenham's desperation was summed up by the late introduction of Tom Huddlestone, who had not played since January.
Spurs' hopes of an equaliser looked to have faded but with three minutes left, Adebayor's shot came back off the post and Sigurdsson converted.
Anichebe had a chance to steal the game for Everton in injury time after he raced through following a mistake by Dawson, but Lloris pulled off a crucial save.