The Spain international became the Premier League champions' most expensive acquisition when a deal worth £37.1 million was sealed on Saturday.
United boss Moyes had been alerted to the possibility the playmaker could be available in January, but did not anticipate luring the former Valencia man to Old Trafford.
Speaking at a press conference to unveil the 25-year-old on Monday, Moyes said: "I'm thrilled. I got wind a month or so ago that it could be a possibility. I never really believed I'd pull it off.
"When we heard Juan would be available we went after him. Congratulations to the club and Ed Woodward (United chief executive) for getting it done."
Mata will wear the number eight shirt and is expected to make his debut against Cardiff City on Tuesday.
The United new boy had been left frustrated after falling out of favour at Chelsea and believes the opportunity to join United will enhance his chances of playing in this year's FIFA World Cup.
He said: "It is really important (to be playing more regularly), because the Spanish squad is difficult to get in.
"We are fighting for positions to get in the last 23 for Brazil, so it is a chance for me to keep playing and keep fit, to show the national manager I want to be there and have to be there.
"I just wanted to go to a place where I could feel happy and I'm happy now. Manchester United is a massive club, it will be amazing for me and it is a massive step."
Mata also believes United can put a disappointing season so far behind them and qualify for the UEFA Champions League by securing a top-four finish.
He added: "What I like most about this club is the character. I think if another club would be in a position like this it would be difficult to come back into a top position, but this club can do it with the character and fighting on the pitch.
"This is the image I have of Manchester United, always fighting with spirit."
