Calvo arrives from Juventus and succeeds Chris Heck, who stepped down at the end of the season. Calvo will take charge from July 1.
Calvo said: “It is a great honour for me to join this Club at such an exciting time. Everyone in football knows the history of and passionate support for Aston Villa.
"In the past few years, we’ve seen this begin to be matched by success on the pitch and progress off it under the ambitious ownership of V Sports.
"I look forward to working with the Chairman, the Board, Unai Emery and everyone at Aston Villa to continue driving this great institution forward.”
Villa co-owner Nassef Sawiris said: “We are pleased to welcome Francesco Calvo to Aston Villa. He has an outstanding track record at some of Europe’s biggest clubs and brings a huge amount of experience.
"Our ambition is for Aston Villa to be competing at the very top level in everything we do, both on and off the pitch. Francesco’s skills, relationships and experience will prove invaluable in making that ambition a reality.”