Alvarez, who featured in his country's run to the World Cup final in Brazil, has made 90 appearances for the San Siro outfit, scoring 14 goals.
And, after agreeing a deal at the Stadium of Light until the end of the season, the 26-year-old expressed his delight at moving to the Premier League.
"I'm very happy to be here. It's a great opportunity, a great club and I'm looking forward to playing in England and playing my part for Sunderland," he told the club's official website.
Alvarez, who started his career at Velez Sarsfield in his homeland, has won seven caps for Argentina.
Coach Gus Poyet has now brought in nine players in this transfer window as he bids to improve on last season's 14th-placed finish.
Sebastian Coates also signed for Sunderland ahead of Monday's transfer deadline.
