The victory means that Ole Gunnar Solskjaer's side have strengthened their grip on second spot in the table and are now nine points clear of fifth-placed Chelsea, while Spurs remain down in seventh position, six points behind fourth-placed West Ham United.
Man United thought that they had made the breakthrough in the 33rd minute when Edinson Cavani swept the ball into the back of the net, but the goal was eventually disallowed for a Scott McTominay foul, with the Scotland international raising his arm into the face of Son Heung-min.
Instead, it was Spurs that took the lead in the latter stages of the first half, with Son firing a low cross from Lucas Moura into the bottom corner from close range.
Both teams had chances in the spell that followed, with Son seeing a low strike saved by Dean Henderson before an extravagant flick from Paul Pogba, which was heading for goal, was blocked by Eric Dier.
The 20-time English champions did take the lead in the 79th minute, though, when a wonderful cross from Mason Greenwood was headed into the back of the net by Cavani.
Greenwood then got his name on the scoresheet in the final exchanges as he worked the space inside the home side's box before smashing the ball past Lloris.
Man United will be looking to book their spot in the semi-finals of the Europa League when they welcome Granada to Old Trafford for the second leg of their quarter-final tie on Thursday night, while Tottenham are next in Premier League action away to Everton on Friday evening.
2nd min: Bright start from Spurs as Son feeds Lucas in a dangerous area, but the Brazilian's effort is blocked behind for a corner by Luke Shaw; the resulting set piece falls to Son on the edge of the box, and the attacker fires over the crossbar.
17th min: Big chance for Man United to score as smart work from Pogba and then Cavani leads to Marcus Rashford striking towards goal, but Dier is across to make an excellent block.
33rd min: Man United appear to make the breakthrough in the 33rd minute of the contest as Pogba delivers a brilliant pass into Cavani, who sweeps the ball into the back of the net through the legs of Lloris.
36th min: The referee disallows Man United's goal, with McTominay, who had already been booked, catching Son in the face with his arm earlier in the move. VAR had signalled for the official to have a look at the monitor, and the play eventually resumes with a Tottenham free kick, leaving the score goalless.
40th min: Tottenham make the breakthrough in the 40th minute of the contest as Lucas delivers a low cross towards the far post, where Son is waiting to sweep the ball into the bottom corner. Victor Lindelof had lost Lucas in the build-up, and it was a smart finish from the South Korea international.
47th min: The final chance of the first period falls to the visitors as Bruno Fernandes delivers a cross towards Cavani, but the ball is just too high for the Uruguay international at the far post.
53rd min: Almost an opportunity for the visitors to level as Fernandes looks for Cavani with a low cross, but Joe Rodon does excellently to clear ahead of the experienced striker.
55th min: Smart save from Lloris to keep out a long-range strike from McTominay, who had found some space in a dangerous position just outside the penalty box.
57th min: Man United level the scores in the 57th minute of the contest as Fred fires home from close range after Lloris had kept out an effort from Cavani.
60th min: Smart save from Henderson to keep out a strike from Son after the ball had broken for the South Korea international in a dangerous area at the far post.
63rd min: Man United come close to a second as Fernandes drives into a dangerous area before striking towards the bottom corner, but Lloris is on hand to make a brilliant save.
67th min: Huge chance for the visitors to score once again as Shaw finds Fernandes, who delivers towards Pogba, but the Frenchman's flick is blocked by Dier.
72nd min: Another smart save from Henderson as Man United's new number one goalkeeper keeps out a near-post strike from Kane with his foot.
73rd min: Aurier goes for a spectacular over-head kick inside the Man United box, but the defender's effort is over the crossbar from close range; replays show that he was offside.
79th min: Sensational second from the visitors as a brilliant cross from Greenwood is headed into the bottom corner by Cavani; it is a goal of the very highest order from the Red Devils.
84th min: The woodwork comes to Man United's rescue on this occasion as Cavani directs it towards his own goal after Pogba had made first contact inside the penalty box.
96th min: Man United have their third of the match as Pogba sweeps the ball wide to Greenwood, who works the space before smashing the ball past Lloris from a dangerous area.
MAN OF THE MATCH - PAUL POGBA
Pogba's performance for the visitors was superb, with the Frenchman always a threat in the final third of the field. The 28-year-old started on the left but drifted all over the pitch and would have had an assist for Cavani's goal had it not been controversially disallowed.
The World Cup winner, who finished with a pass success rate of 88%, should have scored from close range in the second half, but it was still an excellent all-round performance from the midfielder, who set up Greenwood for Man United's third late on.
Referee Chris Kavanagh's decision to disallow Cavani's strike in the first half was controversial to say the least.
The official was sent to the monitor to have a look at an arm from McTominay and decided that it was a foul from the Scotland international on Son, who dropped to the ground.
There did not appear to be much wrong with McTominay's action as he battled to move clear of the Tottenham attacker, but Kavanagh decided that an offence had been committed.
Possession: Tottenham 47%-53% Man United
Shots: Tottenham 12-12 Man United
Shots on target: Tottenham 3-7 Man United
Corners: Tottenham 7-5 Man United
Fouls: Tottenham 11-15 Man United
TOTTENHAM (4-2-3-1): Lloris; Aurier, Rodon, Dier, Reguilon; Ndombele (Lamela 78'), Hojbjerg; Lo Celso (Sissoko 61'), Lucas (Bale 82'), Son; Kane
MAN UNITED (4-2-3-1): Henderson; Wan-Bissaka, Lindelof, Maguire, Shaw; McTominay, Fred; Pogba, Fernandes (Matic 90'), Rashford (Greenwood 71'); Cavani
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