The 30-year-old striker, Villa's record Premier League scorer with 91 goals, joins on a three-year deal.
In six years at Villa, Watkins played a crucial role in the club's rise under manager Unai Emery as they established themselves as regular top-six challengers and won the Europa League, their first European trophy in 44 years.
The England international made himself unavailable for Villa's Premier League opener against Brighton last weekend, which Emery said Watkins apologised for.
"Firstly, I would like to say, never did I intend, nor did I imagine, my time at Aston Villa would end this way," Watkins posted on social media.
"With that being said, I am eternally grateful for the last six years. Not only has it shaped me as a player, but as a person.
"We've been through so many ups and downs in my six seasons here, but one thing that has never changed is that I have given everything for this club."
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