After Manchester City moved into the final Champions League place with a 4-2 win at Stamford Bridge on Saturday, Spurs gave their own hopes a massive boost thanks to first-half goals from Roman Pavlyuchenko and the superb Luka Modric.
Everton, who have beaten Manchester United and Chelsea in recent weeks, looked as though they were still feeling the effects of their UEFA Cup exit at Sporting Lisbon on Thursday night, but responded in a frantic second period through striker Ayegbeni Yakubu.
The race for fourth looks set to go to the wire and Spurs have every chance of realising their Champions League aspirations if they re-produce their performance in the first-half and commitment in the frantic final minutes.
After a lively start, Tottenham took the lead in the 11th minute with their first real opportunity of the match.
Jermain Defoe picked up a lovely long ball from Tom Huddlestone and was given far too much time to fire the ball across goal, where Pavlyuchenko gratefully slid in for his fifth goal in seven days.
While the hosts were dominating possession and territory, Everton could so nearly have levelled on 16 minutes when striker Yakubu capitalised on a dreadful fumble by Spurs goalkeeper Heurelho Gomes.
However, the Nigerian's effort from just a couple of yards out was blocked by Michael Dawson.
Pavlyuchenko and Niko Kranjcar both had efforts saved by Tim Howard before Modric doubled the lead on 27 minutes with a moment of pure class.
The goal was made in Croatia. After some wonderful passing triangles with his compatriots Kranjcar and Vedran Corluka, Modric defied the angles to clip a beautiful shot in off the underside of the crossbar from just inside the area.
The home side looked likely to add a third but Everton manager David Moyes brought on Phil Jagielka for Leon Osman at half-time and his team came out with a new purpose after the break.
And moments after Spurs midfielder Tom Huddlestone was stretchered off following a clash with Johnny Heitinga, Yakubu scrambled the visitors back into the game after Gomes failed to deal with Jack Rodwell's header.
The home crowd clearly sensed panic and as one attack swept after another in the closing stages, the visitors looked increasingly likely to take a point back to Goodison Park.
Rodwell could have equalised almost immediately but Gomes was alert, as he was again when Steven Pienaar ran at the Spurs defence and fired a near-post drive.
And somehow, 13 minutes from time, Everton substitute Landon Donovan missed an open goal from two yards after Rodwell's cross was deflected into the American's path.
Tottenham were nearly undone by a corner once again in the 84th minute but Jagielka's fired his diving header into the stands.
At the other end, Kranjcar brought the best out of Howard after a lung-bursting run from Gareth Bale before Spurs clung on for what could be a vital victory in the final reckoning.