Manchester 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 injury time 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.
It was only the fifth time this season that Alex Ferguson's side have failed to take maximum points from a league game and they have taken just a point from Spurs, who prevailed 3-2 at Old Trafford in September.
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 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.
Defoe shot wide early in the second period, before Bale shaped a beautiful effort with the outside of his left foot that drifted over.
De Gea then 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.
Spurs resumed their pressure, De Gea saving at his near post from Dempsey, before Welbeck narrowly failed to convert a low cross from Rafael at the other end.
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 were destined to leave empty-handed, but with the final whistle beckoning, Dempsey thrashed a loose ball into the bottom-left corner to give United's pursuers a glimmer of hope in the title race.