The Red Devils were forced to come from behind to book a place in their 20th FA Cup final as an early goal from Dele Alli put the Londoners in front, only for an Alexis Sanchez header and Ander Herrera's second-half winner to settle the contest.
Spurs won the ball right from kickoff and claimed the first attack on goal when Harry Kane let fly from just outside the box, his effort taking a deflection on its way over to produce the game's first corner.
Mauricio Pochettino's side took the lead with 11 minutes on the clock as Christian Eriksen played an inviting cross towards the back post for Alli to slide home from close range, sending the Tottenham fans into raptures.
Spurs sought to press home their advantage and almost went two up when Son Heung-min's cross from wide on the right came within inches of Kane's outstretched boot at the far post.
Despite being under the cosh, United drew themselves level in the 23rd minute when Paul Pogba played a measured cross into the centre of the penalty area for Sanchez to meet with a header, the Chilean's effort finding its way beyond Spurs goalkeeper Michel Vorm and into the back of the net.
The Red Devils could have taken the lead two minutes before the interval when Pogba gained possession 25 yards from goal and unleashed a curling strike which forced Vorm to produce acrobatics to tap it behind.
Reacting quickly, Tottenham almost took a lead into the break themselves when Eric Dier struck a low effort from long range, the ball taking a deflection off Chris Smalling before coming back off the post.
The opening stages of the second period were much tighter, and the game would not produce another clear-cut chance until just before the hour mark when Alli's lofted pass picked out Kane inside the box and the England striker let fly, only for Smalling to produce a block to deflect the ball out for a corner.
The match returned to life moments later when Romelu Lukaku lost possession in the final third, allowing Herrera to pick up the pieces by hitting the stray ball with enough power and accuracy to beat Vorm from a tight angle and find the back of the net, making it 2-1 to United.
Jose Mourinho's men nearly added a third in the 64th minute as Pogba's long ball picked out Lukaku's run and the Belgian international let fly, his strike rippling the side-netting.
Spurs refused to sit back and they could have drawn level 10 minutes later when Eriksen unleashed a dipping strike from the edge of the box, which dropped just wide of David De Gea's left post.
Lamela did have a chance to level the match late on when a free kick was floated invitingly into the penalty area, but he could only direct his effort over the crossbar.
United had the opportunity to kill the match off in stoppage time as substitute Marcus Rashford raced clear of the Spurs defence, and would have gone one-on-one with Vorm had Dier not tracked back to make a vital tackle.
The result means that Tottenham's FA Cup semi-final hoodoo continues, with the Whites now having lost their last eight matches at this stage of the competition, and their hopes of winning silverware in 2017-18 are over.
United, meanwhile, will face either Chelsea or Southampton in next month's final.
MANCHESTER UNITED (4-3-3): De Gea; Valencia (Darmian 80'), Jones, Smalling, Young; Matic, Herrera, Pogba; Lingard (Rashford 83'), Lukaku, Sanchez (Fellaini 96')
TOTTENHAM HOTSPUR (4-2-3-1): Vorm; Trippier, Sanchez, Vertonghen, Davies (Lucas 68'); Dier, Dembele (Wanyama 78'); Eriksen, Alli, Son (Lamela 86'); Kane