Adebayor was outstanding and opened the scoring inside the first minute.
Everton levelled in the 15th minute through Phil Jagielka and then took the lead when Kevin Mirallas burst forward and fired home in the 53rd minute.
We responded well to both Everton goals and were camped in the Toffees' half before Sigurdsson finally struck.
Andre Villas-Boas made five changes to the team that started against FC Basel.
Hugo Lloris was back in goal for Brad Friedel while Kyle Walker, Steven Caulker and Michael Dawson started for Kyle Naughton, William Gallas and Benoit Assou-Ekotto respectively. Jan Vertonghen switched to left-back with Caulker partnering Dawson.
Clint Dempsey also returned for Gareth Bale and started on the right of a 4-1-4-1 system.
The game started perfectly as Vertonghen had room to whip in a superb cross from the left that beat Sylvain Distin and was slid home by Adebayor.
The early momentum was with us but Everton had the next chance on 11 minutes when Dawson misplaced a pass out of defence and Leon Osman, spotting Lloris off his line, curled an instant effort just wide.
The Toffees levelled four minutes later when Leighton Baines' corner floated to the far post was met by Jagielka and his downward header somehow squeezed through Lloris' legs at the near post and home.
We recovered quickly from that blow and got back on the front foot. Mousa Dembele, Dempsey, Lewis Holtby and Walker all tried their luck in the next 10 minutes, all missing the target from the 25-yard mark.
Sigurdsson was next up, his curler didn't have enough on it to trouble Tim Howard and Jagielka did well to glance away Dembele's cross from Holtby and Adebayor.
The tempo was high again at the start of the second half and Adebayor almost fashioned a goal from nowhere, holding off two defenders on halfway before surging forward and taking on Jagielka, who timed his challenge well in the box.
However, Everton took the lead on 53 minutes when Seamus Coleman hooked the ball forward from the right touchline to Mirallas, who beat Parker and Caulker before firing across Lloris into the far corner.
We almost responded immediately as Dembele's 20-yarder clipped off John Heitinga but Howard adjusted quickly to flick the ball up and onto the crossbar.
Everton then broke quickly and Lloris had to spring off his line to smother from Osman.
We upped the pressure again. Dempsey fired into the side-netting, Walker burnt Howard's palms after connecting with one of his specials from 30 yards, Parker burst into the box but seemed caught between crossing and shooting and then Adebayor curled straight at Howard.
Tom Huddlestone entered the fray for Dembele on 76 minutes and was soon spraying the ball around as we continued to push forward.
We finally got the reward our efforts deserved in the 87th minute.
Walker darted down the right, his cross was dummied by Parker for Adebayor, who measured his curler from 16 yards only to see it hit the post. The ball rebounded across goal and Sigurdsson was there to finish.
Lloris was forced into a fine late save one-on-one from Victor Anichebe but despite our final efforts in four minutes' added time, we couldn't find the winner.
Spurs (4-1-4-1): Lloris; Walker, Caulker, Dawson, Vertonghen; Parker (Carroll, 87); Dempsey, Dembele (Huddlestone, 76), Holtby, Sigurdsson; Adebayor. Unused subs: Friedel, Assou-Ekotto, Naughton, Ceballos, Livermore.
Everton: Howard; Coleman, Distin, Jagielka, Baines; Gibson, Heitinga; Barkley (Jagielka, 52), Osman, Mirallas (Naismith, 86); Anichebe. Unused subs: Mucha, Oviedo, Hitzlsperger, Stones, Duffy.
Yellow cards: Spurs - Dempsey. Everton - Mirallas.
Goals: Spurs - Adebayor (1), Sigurdsson (87). Everton - Jagielka (15), Mirallas (53),
Referee: Mr A Marriner.
The in-form Iceland midfielder was on the spot to equalise three minutes from time after Emmanuel Adebayor's effort cannoned back off the post.