The French striker latched onto a loose ball 25-yards from goal as full-time approached at Wembley Stadium before striking confidently beyond Everton goalkeeper Joel Robles.
Martial's strike sparked wild celebrations with United players swamped by fans at the side of the pitch.
"It's fantastic," United boss Louis Van Gaal told BBC Sport shortly after the final whistle.
"When you see the joy in the eyes of the players it is fantastic to see. But also for the fans," he added.
United took the lead early in the first half through Marouane Fellaini as the former Everton midfielder coolly slotted away Martial's cutback.
But Van Gaal's side had to ride its luck at times in a game that saw numerous chances wasted.
Everton striker Romelu Lukaku had already spurned a golden opportunity to give Everton the lead before United broke the deadlock. Wayne Rooney raced back to clear the Belgian's shot off the line after he had danced past United keeper David De Gea.
Lukaku was then denied by the feet of De Gea when put through once more as halftime approached.
Everton passed up another glorious opportunity to equalize on 66 minutes when Timothy Fosu-Mensah was harshly adjudged to have fouled Ross Barkley in the United box. Lukaku stepped up but his penalty was saved by David de Gea down low to keeper's right.
At this stage, it appeared highly unlikely to be Everton's day.
But luck finally shone on Roberto Martinez's men when Chris Smalling sliced Gerard Deulofeu's cross into his own net on 76 minutes.
Both side's looked ready to accept extra-time as the 90 minute mark approached.
However, Anthony Martial had other ideas.
"The penalty save was a key moment," De Gea later told BBC Sport. "Everton played really well and had some chances in the end."
"(But) in one second Anthony Martial scores a goal and you win it in the last minute - it's unbelievable," the keeper added.
United will play Crystal Palace in the final at Wembley next month after Alan Pardew's team beat Watford 2-1 in the second semifinal Sunday.
Palace took the lead early when Yannick Bolasie headed home in the sixth minute following a Yohane Cabaye corner.
Fellow EPL side Watford hit back through a Troy Deeney header on 54 minutes, but it was level for just seven minutes.
Pape Souare crossed for Connor Wickham to score the winner, also with a header, Palace holding out to the finish.
It is only the second time, the first in 1990, that Palace has reached the final, but the omens are not so good. It lost to Manchester United on that occasion.