The 20-year-old has sealed his move from Penarol after a deal was agreed last week.
"I'm very pleased to be a part of this club, one of the best teams in the world," Varela, who impressed in a two-week trial with United in May, said on the club's official website.
Varela will represent Uruguay at the Under-20 World Cup in Turkey later this month before joining up with his new team-mates.
He added: "As everyone in the world knows, this is a huge club that has won everything and I really hope that continues.
"I'm very lucky there are quite a few Spanish-speakers here. It will certainly help me settle in."
Varela will provide competition for first-choice right-back Rafael, who enjoyed his best season for the club last term.