Anthony Martial struck a dramatic stoppage-time winner at Wembley after Marouane Fellaini's 34th-minute opener was cancelled out by a Chris Smalling own-goal 15 minutes from time.
United had dominated the first half, with Everton then mounting a comeback during which Toffees striker Romelu Lukaku had a 57th-minute penalty brilliantly saved by David de Gea.
The semi-final looked certain to head into extra time, but Martial then intervened to send the Red Devils into their first FA Cup final in nine years as they remained on course for their first trophy since 2013.
United are currently fifth in the Barclays Premier League and there has been plenty of talk this season about boss Van Gaal being replaced in the summer.
And having praised his team for their efforts on Saturday, the Dutchman - asked how important winning the FA Cup could be for the team's development heading into next season - said: "I am not looking beyond this competition and FA Cup, because we live in the present.
"For the future, I will discuss that with my board and others."
Van Gaal, whose current contract runs until the summer of 2017, was then asked if he would consider winning the FA Cup but failing to qualify for the Champions League a success.
The 64-year-old replied: "It is not for me to judge - it is for the board of Manchester United."
Van Gaal had no doubt his side deserved to win the match.
And although he did not dispute the awarding of the penalty by referee Anthony Taylor following Timothy Fosu-Mensah's challenge on Ross Barkley, he was critical of the official more generally as he gave his assessment of the contest.
"I think in the first half we were the better team," Van Gaal said.
"We scored and created more chances, although we could have finished more.
"In the second half I think the referee changed the match.
"I have to say I think it was a penalty - I don't argue
But I argue about over 50 duels that were not in our favour - when you do that over 10 or 15 minutes in a row, you have a big influence in the game.
"After that we scored the own-goal and it was 1-1, and then you saw we were showing our spirit
It was unbelievable.
"We were out of the game, but then back in it, and then Anthony scores a fantastic goal.
"We are lucky we scored in the last minute but we deserved to win."
For Everton boss Roberto Martinez, the result meant a troubled week got even worse, three days on from his men being thrashed 4-0 by Liverpool in the Merseyside derby.
That was a seventh straight Premier League game without victory for Martinez's 11th-placed side, and he had headed into the semi-final amid growing anger towards him from Toffees fans.
There had also been reports suggesting defeat would spell the end of his tenure.
But after giving his view on what he described as an undeserved and "heartbreaking" loss to United, the Spaniard insisted he was the right man to take Everton forward.
"I am," he said when questioned on the matter
"I've become so attached to Everton, and my ambition is exactly the same as the club's
"I share the drive the club has, which is getting into the Champions League and situations where we can win silverware.
"I have a very strong drive and will give everything I have to fulfil those expectations."
He added: "Every season we are developing and reaching certain signs where I feel we can please our fans and bring the silverware they deserve - but clearly, at moments like this of true disappointment, it is very difficult to see through that the real good things we have developed over the last three seasons."
Source : PA
Source: PA