The Spaniard, who moved to Old Trafford from Chelsea last week for £37.1m, has never won a league title in his career and joins the seventh-placed champions in the midst of a difficult first season under new manager David Moyes.
"I would like to win the Premier League," Mata told MUTV.
"It's a very, very special trophy and I think at this club you can achieve anything. I am young and I have been lucky enough to win some trophies, but I want to win a lot more."
Mata is in line to make his debut for United on Tuesday against Cardiff City.