Teenagers Alejandro Garnacho and Kobbie Mainoo scored to lead United to a 2-1 upset of the holders and Premier League champions as speculation swirled about Ten Hag's future.
Although the Dutchman told ITV he had no doubt he would be back and has said he has received the backing of the club's owners all season, he was less confident in his post-game press conference.
"I don't think about this," Ten Hag said when asked about his future. "I'm in a project and we are exactly where we want to be. We're constructing a team. When I took over it was a mess at United, and we are on our way to construct a team for the future.
"The team is developing, the team is winning and the team also plays to an identity. What you need to play is to be available, you need a strong squad in top football, and especially when you play in England, the Premier League is so competitive."
Garnacho and Mainoo were two of United's bright sparks in a disappointing season in which they finished a worst-ever eighth in the Premier League era.
The teenagers were in fine form at a sun-drenched Wembley and after the final whistle defender Lisandro Martinez hoisted a grinning Ten Hag into the air to celebrate.
"We are delighted for the manager," midfielder Scott McTominay said.
UNBEATEN RUN
Ten Hag, who kissed the trophy before lifting it in front of the United fans, became the first manager to beat Pep Guardiola's City in a major domestic final and ended their 35-game unbeaten run in open play.
"It is a glorious feeling to win the FA Cup final at Wembley," said United co-owner Jim Ratcliffe.
"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."
In Ten Hag's two seasons in charge, United have played in three Wembley finals and lifted two trophies (they won last season's League Cup). If sacking after such a positive finale seems implausible, however, there is precedent.
Louis Van Gaal was fired two days after United's 2-1 FA Cup win over Crystal Palace in 2016 after the team finished fifth in the Premier League.
"We have to keep going, and I'm not satisfied with it, we have to do better and if they don't want me anymore, then I go anywhere else to win trophies, because that is what I did my whole career," Ten Hag said. "That is what they always tell me."
Guardiola had kind words for United's manager.
"(United) have to take a decision. So, I don't know but of course he's a lovely person, an extraordinary manager," Guardiola said.
Ten Hag's press conference ended abruptly when a reporter pointed out that eighth in the Premier League was not good enough for United.
"Sorry to say this, but you don't have any knowledge about football, about managing a football team," Ten Hag said. "When you don't have the players available, then you can't perform, so simple as that.
"And if that is the opinion, it is no problem. Then I go anywhere else and I go and keep winning trophies."
