Ander Herrera was the unlikely United hero, firing a 62nd-minute winner after Alexis Sanchez cancelled out Dele Alli's opener in a pulsating first half.
Spurs pushed for a route back into the game but it was heartbreak for Mauricio Pochettino's men as the club's miserable FA Cup semi-final record extended to eight consecutive defeats.
For Jose Mourinho, who will return to Wembley for the final on May 19, former club Chelsea or Southampton stand between him and the fourth trophy of his tenure at Old Trafford.
Man Utd: De Gea (7), Valencia (6), Jones (6), Smalling (7), Young (6), Matic (6), Herrera (8), Pogba (7), Lingard (7), Sanchez (8), Lukaku (7).
Subs: Darmian (5), Fellaini (n/a), Rashford (5).
Tottenham: Vorm (5), Trippier (5), Sanchez (6), Vertonghen (6), Davies (6), Dier (5), Dembele (5), Eriksen (6), Alli (6), Son (5), Kane (5).
Subs: Wanyama (5), Lamela (n/a), Moura (5).
Man of the Match: Alexis Sanchez
Spurs scored inside the first minute of January's Premier League meeting with United and Harry Kane tried to repeat the feat in a blistering start, only to see a 20-yard shot deflect behind for a corner.
United steadied after the early Spurs onslaught only to fall victim to a clinical 11th-minute attack as Christian Eriksen was played in behind the advanced Ashley Young and crossed to set up Alli for the simplest of tap-ins.
Such was the intensity of Spurs' pressure, United looked on the brink of capitulation, and Kane came within inches of doubling their lead on 14 minutes but he could not stretch enough to connect with Heung-Min Son's deep cross.
United settled again and hit back midway through the half as Paul Pogba outmuscled Mousa Dembele on the left wing, picked out Sanchez with an inch-perfect cross which evaded the Spurs centre-backs, and the Chilean headed home across the stationary Michel Vorm.
Jose Mourinho made five changes following Man Utd’s win a Bournemouth, with Antonio Valencia, Ashley Young, Nemanja Matic, Romelu Lukaku and Alexis Sanchez drafted into the starting line-up. Eric Bailly and Luke Shaw were omitted from the squad completely. Mauricio Pochettino made six changes to the Tottenham side that drew at Brighton. Michel Vorm started in goal instead of Hugo Lloris, with Harry Kane taking the captain’s armband. Kieran Trippier, Davinson Sanchez, Mousa Dembele, Eric Dier and Dele Alli were also recalled.
Just like Spurs, United missed the chance for a quick-fire second as Jesse Lingard diverted a Romelu Lukaku cross over within three minutes of the equaliser.
Pogba's swerving effort worried Vorm in the Tottenham goal on 44 minutes seconds before Eric Dier's drive deflected cannoned off the base of the post.
With the stakes so high, both sides approached the second half with trepidation. It wasn't until the 58th minute that the first clear opening arrived, Chris Smalling throwing his body in the way of Kane's drive.
United continued to ask the greater questions, though, and took the lead for the first time where Herrera arrived from nowhere to apply the finishing touch to a flowing counter involving Sanchez and Lukaku.
United, aided by a raft of lengthy substitutions, then saw out the victory to reach another FA Cup final, Spurs restricted to shots from distance as their bid for a first trophy in a decade fell short.
- Manchester United have reached their 29th major domestic cup final (20 FA Cup, 9 League Cup), more than any other side in English football.
- United have reached the FA Cup final in two of the last three seasons, having failed to reach the showpiece in eight consecutive seasons previously.
- Tottenham haven't lost a single Premier League game in which they've led this season, but have now been eliminated from their three other competitions this season in games in which they took the lead (2-3 vs West Ham in the League Cup, 1-2 vs Juventus in the Champions League).
- Spurs also extended their record run of consecutive FA Cup semi-final defeats to eight, last winning one in 1991.
- Alexis Sanchez has scored eight goals in his eight appearances at Wembley for club and country, including six goals in five FA Cup matches.
- Ander Herrera has scored in each of his last two FA Cup games, as many as he had in his previous 15 in the competition.
- Jose Mourinho has reached his sixth major domestic final in England (2 FA Cup, 4 League Cup), winning each of his previous five.
Jose Mourinho: "The players deserved to win. When you deserve [to win], especially against teams like Spurs, it's even more important. Both teams started very strong, very aggressive. They were better than us, they scored and there was a period when we lost control in midfield. But the second half was quite calm because we had the feeling it would be very difficult to lose the game and most probably we would win it."
Mauricio Pochettino: "I have no regrets. We competed and dominated in the first half. It was unfair to go in 1-1 at half-time. But in the second half they scored the game was competitive. When you compete against teams like Manchester United it's always 50-50, and today the victory was for them."
Man of the Match - Alexis Sanchez
It's never easy for January signings to bed in at their new clubs but even the most-ardent Manchester United supporter would admit Alexis Sanchez's start to life at Old Trafford has been underwhelming. During his Arsenal career, the Chilean often reserved his best for games against Spurs or at Wembley, and a combination of the two saw him rekindle his form of old. Having hauled his struggling side level, Sanchez set the tone from there on in what was arguably his most influential performance in a United shirt yet.
Manchester United host Arsenal at Old Trafford on Nissan Super Sundaybefore Tottenham entertain Watford at Wembley on Monday Night Football.