The 22-year-old, who was captain of the League One side, says that he is looking forward to playing in the Premier League and has compared himself to Michael Carrick.
"I see myself as a Michael Carrick-type of player," said Westwood. "Someone who keeps it simple and retains possession.
"From the conversations I've had with the manager, I think that I will fit in perfectly here.
"This is a fantastic club, a massive club and I can't wait to get started."
Villa boss Paul Lambert paid tribute to his new signing.
Lambert said: "Being captain of his football club at 21, I think that says a lot about Ashley's quality, the responsibility he has taken and his temperament.
"He's also a terrific prospect for this football club - a real, real talent."
Westwood, a product of the Crewe youth academy, is Villa's seventh signing of the summer.