On an Arctic-like afternoon in north London [ Images ] Robin van Persie's [ Images ] 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.
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 [ Images ] 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 [ Images ] 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.
Chelsea hang on to beat Arsenal
Chelsea enjoyed some much-needed home comfort with goals from Juan Mata and Frank Lampard [ Images ] securing a 2-1 victory over Arsenal in the Premier League on Sunday.
Arsenal staged a second-half revival with Theo Walcott [ Images ] halving the deficit at a snowy Stamford Bridge [ Images ] but Chelsea hung on for only their second win in six home league games since Rafa Benitez [ Images ] was hired as interim manager.
Victory lifted third-placed Chelsea to 45 points from 23 games, 10 behind leaders Manchester United whose game at Tottenham Hotspur will go ahead later in north London despite heavy snow putting the fixture in doubt.
Mata scored after six minutes after Cesar Azpilicueta's lofted ball put him through on goal, although Arsenal were seething at the referee's failure to award a free-kick their way in the initial stages of the move.
Lampard made it 2-0 from the penalty spot after 16 minutes after Arsenal keeper Wojciech Szczesny escaped a red card for bringing down Ramires in the area.
The midfielder's calm finish was his 195th goal for the club and he is now just seven behind Chelsea's all-time record goalscorer Bobby Tambling.
Walcott's superb finish after a pass from Santi Cazorla just before the hour gave Arsenal hope and Chelsea endured a nervy final half hour as the Gunners pressed for an equaliser.
Chelsea survived some anxious moments although at the other end Fernando Torres [ Images ] and Demba Ba, who was left on the bench, wasted chances to wrap up the points.
Arsenal manager Arsene Wenger [ Images ], whose side now face a scrap to achieve a top-four finish, said a poor first half and some dubious officiating had undermined his side.
"I thought Francis Coquelin was injured for the first, and I'm not convinced at all that it was a penalty on the second goal," Wenger told Sky Sports.
"We made some changes at halftime and came out with a different attitude with much more initiative. We had Chelsea on the back foot but could not get it back to 2-2."
The victory was Chelsea's first at home this year and lifted some pressure on Benitez who was booed in midweek when Chelsea were held 2-2 at home by Southampton.
"We have been doing well, creating chances in all our games. In the games we didn't win we deserved to win. It was just a matter of time until we had a performance like this," Benitez said, who was full of praise for Torres, said.
Photograph: Clive Mason/Getty Images