The Benin international striker had a medical at Villa Park before finalising the move for an undisclosed fee, after reported interest from Watford and Norwich City.
He becomes Villa’s second capture of the day, after the arrival of Jordan Veretout from Nantes.
Gestede has attracted interest from a host of clubs after scoring 22 goals in 44 games for Rovers last season.
The 26-year-old was close to leaving the Championship side in January with Crystal Palace having a bid rejected.
Hull City and West Brom were put off by the £6m price tag, according to Sky sources, but Villa boss Tim Sherwood snapped up one of his top targets.
“I’ve known about Rudy since he was at Cardiff City and he is a player I’ve always admired,” Sherwood told the club’s official website.
“His goalscoring ratio last season was fantastic and he is someone who certainly knows how to find the back of the net. He’s a big, physical striker but he can play and I’m pleased to welcome him to the football club.”
Villa be hoping the French forward can provide the firepower to replace Christian Benteke, who joined Liverpool for £32.5m last week.
Sherwood’s side have also been linked with a move for Tottenham striker Emmanuel Adebayor, and the Togo striker is expected to move to the Midlands on a season-long loan deal.
There was further good news for Villa fans as the club announced that defender Nathan Baker had signed a new four-year contract.
The 24-year-old academy product, who has played 87 times for the club since his first-team debut in 2011, said: “I am delighted to sign a new contract here at Villa. I am determined to impress the gaffer and get in that side.
“The competition is going to be fierce here at centre-half but that’s what the gaffer wants. There’s a lot of choice for him. But you can’t do well as a club without competition for places.”