United will face Crystal Palace or Watford in a 19th FA Cup final after Anthony Martial's late, late winner at Wembley.
Marouane Fellaini had put Louis van Gaal's side ahead in a dominant first half but - after David de Gea saved a Romelu Lukaku penalty - Chris Smalling's own goal pegged the Reds back and looked set to have forced extra-time.
However Martial latched on to substitute Ander Herrera's through ball to take a touch and slide a composed finish past Joel Robles in the third minute of stoppage time to send United's travelling supporters into ecstasy.
United created the majority of the early openings but, in Lukaku, Everton had a pacy and powerful outlet and it was the Belgian who went closest after ten minutes.
Wayne Rooney, playing in a deep midfield role, ran the length of the field to head off the line after Lukaku had rounded De Gea and clipped goalwards from a tight angle.
Marcos Rojo headed wide, Jesse Lingard had a shot saved by the sprawling Robles and Rashford also went close from Fellaini's knock down in an increasingly frenetic contest.
Martial should have done better when put through by Lingard, but the Reds did not rue that miss and deservedly went ahead eleven minutes shy of the interval.
Martial burst towards the touchline, skipped past Muhammad Besic and pulled the ball back for Fellaini to steer home from close range against his former team.
The second half began as the first had ended- with United on top and creating more chances.
The best of them fell to Lingard after a move involving fellow young players Tim Fosu-Mensah, Marcus Rashford and Martial, whose backheel set up Lingard but he fired wastefully wide.
Everton belatedly sparked into life and were given the chance to level the match when Aaron Lennon found Barkley whom went down under a challenge from Fosu-Mensah and referee Anthony Taylor pointed to the spot.
De Gea dived brilliantly to his right to palm Lukaku's spot-kick away and United continued to be indebted to the Spaniard as Martinez's men then came on strong.
Barkley lashed an effort wide, former United man Tom Cleverley was kept out by De Gea whom then thwarted substitute Gerard Deulofeu with a superb stop.
Everton continued to press and finally got a foothold when Deulofeu's cross was inadvertently turned into his own net by the stretching Smalling with 15 minutes to play.
It looked like extra-time would be needed, but the Reds never-say-die spirit made sure that would not be the case, as, three minutes into added time, Martial latched on to Herrera's poked through ball to send his side into next month's showpiece.
Seen you soon, Wembley!
Overall team performance: 8/10
United Faithful Man of the Match: Marouane Fellaini. Scored the opener, excelled in defence and attack and made an outstanding tackle on Lukaku at 1-1.