City captain Ilkay Gundogan scored the fastest goal in an FA Cup final, with just 12 seconds on the clock at Wembley.
Bruno Fernandes levelled from the penalty spot but Gundogan struck again early in the second half and the game ended 2-1.
“We are broken, disappointed of course, but I said I am proud of my team,” Ten Hag told the BBC.
“We did very well, we conceded two soft goals but we were in the game. There was a great spirit and very good organisation.”
He added: “This team showed resilience, character and personality. We know we have a way to go but this will make us better. It was a test for us — we didn’t succeed but we can take a lot of positives into next season.”
United captain Fernandes said United had made “big steps for next season” despite the defeat.
The Red Devils broke their six-year trophy drought in February, beating Newcastle in the League Cup final, and finished third in the Premier League in Ten Hag’s first season at the helm.
But they were second best in the London sunshine as Premier League champions City remained on course for a treble.
“Everyone is down and it’s really difficult,” said Fernandes. “We wanted to end the season in a different way but it wasn’t possible.
“We’ve made big steps for next season. It wasn’t a successful season overall but it was a good one. We have to come back next season and try to win bigger trophies.”
The Portugal midfielder said United’s supporters deserved more.
“Our fans have been great all season and they deserved a different result today,” he said.
“We’re disappointed that we couldn’t give the fans the joy they deserve.
“We are grateful for everything they have done this season. I promise we will come and try again next season.”
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