Leicester's earlier 2-0 win at relegation struggler Sunderland temporarily opened up a 10-point gap but Tottenham responded in some style with its first home league win over United in 15 years.
"The performance was perfect in the second half," Tottenham manager Mauricio Pochettino said.
The visitor was caught in heavy London traffic on the way to White Hart Lane, reaching the ground with only 15 minutes to the scheduled kick off, which was delayed for half an hour.
United manager Louis van Gaal was clearly unhappy with the situation, but despite his team's preparation being disrupted it made the stronger start in a key fixture for both.
But Spurs gradually found its feet and carved out the clearer chances as the half drew to a close.
Erik Lamela had the best, heading wide with only David de Gea to beat and at halftime it was goalless.
Anthony Martial worked United's best opening for himself just after the hour which Hugo Lloris pushed away, but soon the home side was ahead.
Harry Kane and Lamela won a midfield tussle to send Christian Eriksen clear on the left. The Dane quickly found Dele Alli, who converted for his eighth of a memorable season the day before he turned 20th.
Lamela, fully atoning for his first half miss, then picked out Toby Alderweireld with a free kick and the defender headed home the second.
Lamela's afternoon was complete as he completed a spell of three goals in six minutes by converting Danny Rose's low cross.
"We knew how good the opposition were and it was important that we keep on winning games," England midfielder Alli told Sky Sports, but it has only five games to make up the deficit.
The defeat dealt a serious blow to United's hopes of a top-four Champions League qualifying spot, four points adrift of city rival Manchester City, with both having six games left.
By contrast, Leicester's fairytale continues, with the later result ensuring it will at least finish in that top four and play in Europe's top club competition for the first time.
That would be the very least it could expect, just three wins from the title for the 5,000-1 outsider at the start of the season.
The Foxes' Italian manager Claudio Ranieri was close to tears as he spoke about its win at the Stadium of Light, sealed by two Jamie Vardy goals in the second half..
It was a typically resolute performance from Leicester, "solid" according to Ranieri, but sealed a fifth consecutive EPL victory.
"You make this job for the emotion you feel inside but it is difficult for me to tell what kind of emotion," he told BBC Sport.
Sunderland, still four points adrift of safety, pressed hard for an opener, but fell behind to a trademark Leicester goal, Danny Drinkwater finding England striker Vardy to race clear to score in the 66th minute.
Jack Rodwell wasted a golden chance to level for the Black Cats before Vardy stole way again for his 21st league goal this season and first from open play in the league for two months.
"It is important for Jamie Vardy to score because he made some good assists in the last game but he is our goalscorer and he needed to score again and I am very happy with him," Ranieri added.
In Sunday's other game, Liverpool warmed up for its crucial Europa League quarterfinal home tie against Borussia Dortmund on Thursday with a 4-1 thumping of Stoke City.
Dortmund was held 2-2 Sunday at Schalke in the Ruhr derby, leaving it seven points adrift of Bayern Munich in the Bundesliga title race.
With just five games left, Bayern is headed for a historic fourth straight title
"It is very likely that Bayern will be champions," said Dortmund coach Thomas Tuchel, whose side twice led at the Veltins-Arena.