United had taken the lead through Robin van Persie in the 25th minute and looked set to hold on for a narrow victory until Dempsey scrambled the ball home from close range in the 93rd minute at a snow-sprinkled White Hart Lane.
The result means that second-place Manchester City are now five points behind United, while Spurs finish the weekend trailing third-place Chelsea by four points and leading Everton by the same distance.
“We did so well at everything and we kept heading it out — the disappointment is not finishing them off,” said United manager Alex Ferguson.
“We had a lot of opportunities on the counter-attack but our final ball let us down. Tottenham were very committed and aggressive. I think they have a decent squad and a good team.”
It was only the fifth time this season that Ferguson’s side have failed to pick up maximum points from a league game and they have taken just a point from Spurs, who prevailed 3-2 at Old Trafford in September.
“It was a big relief. I thought we dominated the game,” said Spurs manager Andre-Villas Boas.
“We wanted to win, but it was an important point in the end due to the form United are in. A point can give us a lot of confidence.”
Persistent snowfall in north London had put the fixture in jeopardy, but the game survived a pitch inspection 70 minutes prior to kick-off and as the game got under way, the snow had started to ease.
Spurs looked to take the initiative, Dempsey seeing a volley blocked by Nemanja Vidic and the recalled Scott Parker shooting over from 22 yards, but United threatened first when van Persie headed straight at Hugo Lloris.
Few players have caused Patrice Evra problems as consistently as Aaron Lennon in recent seasons and the Spurs winger left the Frenchman trailing in the 14th minute before drawing a save from David de Gea with a low shot.
United started the game with Wayne Rooney on the bench but there is always danger when van Persie is around and in the 25th minute the Dutchman struck for the 10th time in as many league games to put the visitors ahead.
Danny Welbeck drifted in from the left before spraying the ball wide to England colleague Tom Cleverley, and his curling cross was headed in at the back post by the leaping van Persie.
Spurs resumed their assault on the United goal but both Jermain Defoe and Gareth Bale were thwarted by agile saves from De Gea, who readjusted well to block the Welshman’s deflected shot shortly before half-time.
Early in the second period, De Gea produced an incredible reaction save with his legs to thwart Dempsey, who was able to take aim at an exposed goal from just eight yards after a slalom into the box by Mousa Dembele.
United manager Ferguson introduced Rooney with half an hour remaining and he immediately had a penalty appeal rejected after being caught by Steven Caulker just inside the Tottenham area.
A clear sight of goal arrived for the hosts in the 78th minute, but when Lennon sent Defoe clear, his former West Ham United team-mate Rio Ferdinand slid in from nowhere to prevent an equaliser with a vital block.
Ferdinand then produced a telling intervention with five minutes left, bravely getting his head to a fierce Bale strike to divert the ball narrowly wide of the left-hand post.
It seemed Spurs’ efforts would be in vain, but with the final whistle beckoning, De Gea miscued a punch and Dempsey turned Lennon’s pass home to give United’s pursuers a glimmer of hope in the title race.