Engels becomes Villa's latest addition for a fee believed to be around £7 million ($9m).
Villa are set to return to the Premier League this upcoming season after attaining promotion from the Championship and have spent big money to ensure the team is ready to compete and avoid a quick relegation.
The club has now spent nearly £100 million to retool the squad ahead of the return to the top flight, with Engels the latest signing to be part of that project as the 24-year-old defender joins from Stade Reims.
The defender spent last campaign on loan with the Ligue 1 club, where he made 33 appearances before the French side opted to trigger a purchase option in Engels’ contract.
"Bjorn is a player that I've been watching for the past 18 months," said Villa boss Dean Smith.
"We're very happy to have secured his services and look forward to working with him."
Vila have spent more money than any other Premier League team this summer after losing a dozen players at the end of the season.
The club's biggest move came in the form of Brazilian striker Wesley, who cost €24 million (£22m/$27m) as Villa added the forward from Club Brugge.
Joining him are a number of defensive options, with Tyrone Mings, Ezri Konsa, Matt Targett and Kortney Hause also signing on with the club.
Villa secured their place in the Premier League via the promotion play-off, where they topped Derby County in the finale to book their spot in the top flight.
They're joined by Norwich and Sheffield United to complete the trio of Premier League newcomers.