The 35-year-old, the Republic of Ireland's most-capped international, on Monday night signed a five-year contract to become McLeish's first signing at Villa Park. He had been frozen out under City boss Roberto Mancini as England international Joe Hart made the position between the posts his own.
"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."
The signing of Given is expected to be followed by further arrivals as McLeish looks to find replacements for Ashley Young and Stewart Downing, who have made big-money moves to Manchester United and Liverpool respectively.
The club have already had a bid for Wigan winger Charles N'Zogbia refused but are expected to go back with an improved offer while they have also been heavily linked with Tottenham defender Alan Hutton and West Ham's England midfielder Scott Parker.