The 20-year-old joins the Premier League champions from Uruguayan side Atletico Penarol for an undisclosed fee, having impressed during a two-week trial period at Carrington.
The defender will link up with United after representing Uruguay at the U20 World Cup in Turkey later this month.
"I'm very pleased to be a part of this club, one of the best teams in the world," said Varela.
"As everyone in the world knows, this is a huge club that has won everything and I really hope that continues."
He added: "I'm very lucky there are quite a few Spanish-speakers here. It will certainly help me settle in."
Varela becomes the first signing for United since David Moyes replaced Sir Alex Ferguson as manager at Old Trafford.
Sky Sports revealed earlier this week that Varela was set to finalise a move to Old Trafford.