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On an Arctic-like afternoon in north London Robin van Persie's first-half goal appeared to have secured the points for United until Dempsey slid home an equaliser with virtually the final kick of the game.
The leaders have 56 points from 23 games with Manchester City on 51 and Chelsea, who beat Arsenal 2-1 earlier in the snow-hit capital, on 45, four ahead of Tottenham.
Dutchman van Persie, often a thorn in Tottenham's side when he was at Arsenal, took his league tally for the season to 18 with a header from Tom Cleverley's cross midway through the opening period of an absorbing game.
Tottenham, whose strong recent run had seen them establish themselves in the top four, were twice denied equalisers by the outstretched legs of United keeper David De Gea as they attempted a rare league double over Alex Ferguson's side.
Gareth Bale thought he had equalised shortly before halftime but De Gea performed marvels to keep out his deflected shot and after the break Dempsey wriggled through only to see the keeper again come to United's rescue.
United, who began with Wayne Rooney on the bench, showed another side to their character. With resolute defending rather than fluent attacking play, they were content to absorb pressure and the policy seemed to have earned an important victory.
As the snow poured down, Bale also went agonisingly close to a late equaliser with a shot deflected past the post.
However, the hosts kept plugging away and Dempsey was on hand to equalise after De Gea could only punch a clearance to Aaron Lennon and the winger threaded a pass to the American.