Given, 35, is the Republic of Ireland's most capped player club and goes straight into the squad for their next friendly, against Walsall on Thursday.
The former Blackburn and Newcastle keeper moved to City in 2009 for £7million but found first-team opportunities limited last season after England's Joe Hart became Roberto Mancini's preferred option.
Given started his career as a youngster at Celtic and had been heavily linked with a summer move to Parkhead, but McLeish has now secured the veteran's signature to replace the outgoing Brad Friedel, who joined Tottenham at the start of the month.
Given told Villa's official website he cannot wait to get started.
He said: "Aston Villa is a huge football club and I'm really excited to be joining. I'm looking forward to training and playing with the players here. They're a good bunch of lads, very talented and I know several of them already which will help.
"Everyone knows that I didn't play enough football last season, so I'm really pleased that I'll be able to put that right here. Hopefully, this move will get me back to where I was before in the Premier League, which is where every player wants to be playing.
"Obviously, I'm also excited to be working with Alex McLeish and it's going to be a fresh start for everybody at the club this season.
"It's also a new challenge and one I'm eager to embrace.
"My aim is to impress the Aston Villa supporters with my performances and I can't wait really to get started."
McLeish, who has already seen Friedel, Ashley Young and Stewart Downing leave the club since his controversial appointment in June, believes his first signing as Villa boss is "world class".
"Shay has come here probably with something of a point to prove and that won't do any harm at all," McLeish told www.avfc.co.uk.
"Everybody who has watched him perform over many years in the Premier League will know his pedigree, so we believe we have secured the services of one of the best goalkeepers in the business. He's a real competitor and a world-class keeper and I'm delighted he's the first signing I've made at Villa."