The 20-year-old signed a five-year contract after impressing the Premier League champions in a two-week trial last month.
Varela, who is expected to play for Uruguay at this month's Under-20 World Cup, said he could not wait to get started.
"I'm very pleased to be a part of this club, one of the best teams in the world," he told United's official website.
"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," he said. "It will certainly help me settle in."