Goals by Marcus Rashford and Ander Herrera sank Chelsea and, with six games left for the top two, there is little room for any more slip-ups by Antonio Conte's side who a few weeks ago looked like they were running away with the title.
While Chelsea remain favourites to win the league, Conte will be concerned about the manner of the defeat at Old Trafford where they were rarely able to turn possession into chances.
The west London side are top with 75 points, ahead of Spurs on 71, also from 32 games, Liverpool with 66 from 33, Manchester City on 64 from 32 and United with 60 from 31.
"Manchester United deserved to win because they showed more desire, more motivation, more ambition to win this game," Conte told the BBC.
"This league is not closed. Tottenham are on great form and playing with great enthusiasm but we are playing a great season and we must try to reach this target."
The strikes by Rashford and Herrera at the start of each half were enough for United who, despite resting several players against Chelsea ahead of Thursday's Europa League quarter-final return leg at home to Anderlecht, were good value for their win.
United manager Jose Mourinho started with top scorer Zlatan Ibrahimovic on the bench but delivered a tactical masterclass against his former club, largely nullifying the threat from the usually dangerous Eden Hazard.
"Chelsea are the best counter-attacking team in the country and we controlled them very well. I'm really happy with the boys. I'm happy - not because it's Chelsea, it's because we need these three points," Mourinho told the BBC.
"I don't feel extra joy at beating Chelsea - we beat the leader. It doesn't matter if the leader is Chelsea or another - we beat them convincingly."
While United's hopes of making the top four will be boosted by the win, the major impact will be felt in north London where Spurs will sense a chance to chase down Chelsea's four-point lead in the final six games of the season.
Earlier, Liverpool reclaimed third spot in the Premier League after Roberto Firmino's header sealed a hard-fought 1-0 victory against West Bromwich Albion.
West Brom created the best chances in a largely uneventful first half before Firmino nodded home Lucas' flick on to register his 11th league goal of the season just before the break.