Eden Hazard dazzled during the early stages and the Belgium forward won and converted a 22nd-minute penalty.
United pushed Chelsea back after the break and Paul Pogba missed a glorious chance eight minutes from time.
But there was no way through, leaving Mourinho to embrace his rival Antonio Conte at full-time before reflecting on a game he viewed as a hard luck story.
"I congratulated them because they won but I do not think they deserved to win," he told BBC Sport.
"I congratulate them because I am a sportsman. They scored one more goal than us and they get the cup, and I have to do what my job asks me to do. I am Manchester United manager and I have to be respectful.
"I think we deserved to win, we were the best team. But that's football.
"I'm quite curious to know what people say and people will write because I can imagine if my team played how Chelsea did what people will say."
Romelu Lukaku declared himself fit only for place on the bench on his return from an ankle injury and Mourinho felt United missed the Belgium striker's presence in attack.
"I gave everything I could, the players gave absolutely everything. It was a bit hard for us to play without Lukaku against a team who defended with nine players," he said.
"You need that presence in the box and we didn’t have it for a long time
"Every defeat hurts but I go home with the feeling that we gave everything, so no regrets."
Mourinho added: "We knew that this is the way they normally play, with a little bit more ambition in their counter-attack, but after 1-0 they were really defensive.
"They are not stupid, they know our team without Lukaku or [Marouane] Fellaini, we don't have a presence so they knew they could have dominance in direct football.
"Courtois had a couple of saves, we were unlucky with some balls in the box - Paul [Pogba]'s header is one. I am a sportsman, the team that scores one more goal wins and that is why I congratulate them."