The 20-year-old joins the Premier League champions from Atletico Penarol for an undisclosed fee and will now link up with his national side for the Under-20 World Cup.
Varela made just one senior appearance for Penarol but impressed during a trial spell at Old Trafford at the end of last season.
It is reported the right-back has cost the club £1 million.
A club statement confirmed: "Manchester United is pleased to announce it has reached agreement to sign 20-year-old defender Guillermo Varela from Uruguayan club Club Atlético Penarol for an undisclosed fee.
"Guillermo will immediately join up with the Uruguay Under-20 squad for the upcoming FIFA Under 20 World Championships in Turkey."