The Premier League side have reportedly paid £14m for the French midfielder, with the deal potentially rising to £15.5m with add ons.
Sanson has signed a four-and-a-half-year deal that will keep him at Villa Park until 2025.
"I'm really happy to be here. I'm very proud to join the Aston Villa family, and in the future I hope we can do great things together," Sanson told Villa's website.
"It's a magical club, a big club in England. When Aston Villa called me, I didn't have to think about it for a long time – it was the best opportunity for me and my career.
"It was a dream for me to join the Premier League since I was a little boy. I've seen the games and I like the intensity in the Premier League. For me, it was a dream and I'm very proud to join the Premier League with Aston Villa."