Starting the match 10 points adrift of leaders Leicester after they had beaten Sunderland earlier, Tottenham shaved the gap to seven with Dele Alli, Toby Alderweireld and Erik Lamela on target to earn the north London club a first home league win over United since 2001.
With five games remaining second-placed Tottenham, chasing a first title since 1961, have 65 points to Leicester's 72 while fifth-placed United's hopes of ending a disappointing season in the Champions League places are receding.
"The manager always wants a performance from us and playing the right way and we did that," Tottenham's England midfielder Eric Dier told Sky Sports.
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"We will keep on fighting until the end. That's what everyone expects but credit to Leicester."
The match kicked off half and hour late after United's team coach was stuck in traffic and the atmosphere was flat in a first half in which Lamela wasted the best chance when he headed Christian Eriksen's pass wastefully wide.
Tottenham began the second half with much more urgency although they were given a scare on the hour when Anthony Martial cut in from the left and stung Spurs keeper Hugo Lloris's gloves with a powerful drive.
That was United's only shot on target but it simply spurred on the hosts who took the lead in the 70th minute when Harry Kane released Eriksen and his early ball across the area was met by Alli who beat David de Gea.
Alderweireld climbed high to head in Lamela's free kick four minutes later and with Spurs on the rampage, Lamela then stroked home Danny Rose's low cross to the delight of the home crowd who still believe their side can reel in Leicester.
Fifth-placed United are now four points behind Manchester City.
(Reporting by Martyn Herman, editing by Pritha Sarkar)
