Robin van Persie had looked to have claimed all three points with his first-half header at White Hart Lane on Sunday but Dempsey swept the ball home in stoppage time to salvage a point.
The adverse weather conditions had threatened to see the Premier League clash at White Hart Lane called off - and Spurs fans were soon left wishing it had.
After a cautious start in which Aaron Lennon forced a save from David de Gea, it was ex-Arsenal man Van Persie who opened the scoring by heading home Tom Cleverley's right-wing cross.
It was the culmination of a flowing move from the Red Devils after good work from Michael Carrick, Shinji Kagawa and Danny Welbeck, who was selected ahead of Wayne Rooney in attack.
The home side struggled to create outright chances until De Gea produced a superb reaction stop with his legs to deny Gareth Bale's deflected effort from the edge of the box just before the break.
And De Gea was called into action once again after the interval when he came up with a reflex save to keep out the advancing Dempsey from point-blank range.
The Spanish goalkeeper then pushed away another Dempsey shot shortly afterwards as Spurs pushed hard for an equaliser and the crowd got behind their team.
But United remained a threat on the break and Rooney had a strong shout for a penalty shortly after replacing Kagawa when he went down under a challenge from Steven Caulker.
It was still Andre Villas-Boas' side asking most of the questions and it required a last-ditch block from Rio Ferdinand to prevent Jermain Defoe levelling things up after getting away in the channel.
But just when Sir Alex Ferguson's team looked to have held on for the win, De Gea could only punch a cross as far as Lennon inside the area and he returned the ball to Dempsey to stun United.