Former Penarol fullback Varela won a contract at United after successful trials.
"I'm very pleased to be a part of this club, one of the best teams in the world. I'm very happy," he said. "This is a huge club that has won everything and I hope that continues."
Varela, who says pace is his best asset, spent a fortnight on trial with the Reds at the end of last season, when he got to know fellow Spanish-speaker Javier Hernandez.
"He was a great help because he translated everything that was being said in English into Spanish for me while I was here," Varela added. "I also got to know most of the squad."