Fish joins Cardiff for £1m plus £3m in bonuses. He's penned a four-year contract. United also retain 30 per cent sell-on clause.
Fish said: "It’s a huge club. The direction that the club want to go in, you’ve seen a lot of good signings in this window.
"Being a part of that is really exciting and I’m happy to be here. Hopefully I can do my best for the team."
Cardiff coach Erol Bulut said: “He is a young defender, who has been at Manchester United for many years, so has had a good education from them.
"For the last two years he was in Hibernian, so we watched him and we were interested in him in January too.
“We hope that he will progress with us in a good way. He is still 21 years old, just starting his career, and he is good on the ball, good in duels, so he will add to our playing style."