Teenage stars Alejandro Garnacho and Kobbie Mainoo struck in the first half at Wembley and United then dug deep to win 2-1 after City substitute Jeremy Doku's late effort.
It means Ten Hag's side finished a miserable season on a high and also qualified for next season's Europa League.
The build-up to the second successive all-Manchester FA Cup final was dominated by talk over Ten Hag's future after a report on Friday said he would be sacked by United's new co-owner Jim Ratcliffe regardless of the result.
United finished eighth in the Premier League -- their lowest placing since 1990 -- and crashed out of the Champions League at the group stage.
As fans celebrated wildly in the London sunshine, Ten Hag waved to them while clutching the FA Cup, but it was impossible to tell whether he was saying goodbye.
The Dutchman, at the end of his second season in charge at Old Trafford, admitted he still did not know what the future held.
"I don't know (if I will stay)," he told the BBC.
"The only thing I am doing is preparing my team, developing my team, progressing my team and individual players. This is for me a project.
"When I came in things were a mess. We are now better but by far not where we want to be.
"Football is about winning trophies. I want to play the best football, I want to play dynamic football, but in the end you have to win trophies. I am very proud of the players and staff because they did an incredible job."
Ratcliffe, whose INEOS Group is now in charge of football operations at Old Trafford, said he was proud of the players but did not comment on Ten Hag's position.
"It is a glorious feeling to win the FA Cup final at Wembley," he said. "Manchester United clearly were not the favourites to win today but they played with total commitment and skill and overcame one of the great teams in football.
"We are all very proud of the players and the staff who work tirelessly to support them."
Harsh criticism
Ten Hag, 54, said United had shown what they could do when they had key players available following an injury-ravaged season -- with Lisandro Martinez alongside Raphael Varane at the centre of defence.
"I tell you this all year -- when the players are fit we can play good football," he said. "A very good performance against the best team in the world.
"With a local rivalry it's there and it's much more important when you play a team from your own city. But for us it wasn't about this, it was to prove a point after so many setbacks. The team showed so much resilience and I'm proud of them."
He said he felt criticism of him and his team had been unfair despite their troubles.
"We didn't have the players," he said. "We have seen the same things, not always good football, definitely not, but we had to make compromises all the time and then you can't play the football you want to play.
"I had maybe three or four times in two years the full squad available. Even today we were missing massive players like Harry Maguire, Luke Shaw and Casemiro."
United captain Bruno Fernandes said: "It is crucial for everyone. We know the manager is under so much scrutiny, he deserves this.
"Also everyone in the backroom staff and the players, we all deserve this."
