Lambert tendered his resignation to the Canaries on Thursday amid reports Villa had made an official approach for his services.
His resignation was not accepted but an agreement has now been reached for the Scot to make the switch to the West Midlands.
A brief statement on Villa's website read: "The board of Aston Villa are delighted to confirm that Paul Lambert has been appointed Villa manager."
It is believed Villa will have paid around £1million in compensation to ensure Lambert becomes their fourth manager in less than two years.
Lambert will be charged by club owner Randy Lerner with the task of revitalising Villa after two seasons in the doldrums. After three successive top-six finishes under Martin O'Neill, Villa have dropped too close to the Barclays Premier League bottom three for comfort.
Gerard Houllier replaced O'Neill but had to quit last summer on health grounds, while former Birmingham boss Alex McLeish was never accepted by the Villa fans. Lambert fits the profile of being a relatively young, hungry and ambitious manager after working his way through the lower leagues before transforming Norwich's fortunes.
That is certainly the feeling of Villa midfielder Barry Bannan, who said before Lambert's appointment: "He has done very well with Norwich and the clubs he was at before then. He has a good record up until now so hopefully if it is Lambert he can be as successful with us as he has with his previous teams."
Lerner also held talks with former Manchester United star Ole Gunnar Solskjaer but he opted to stay at Molde in Norway for family reasons, although he was not formally offered the Villa post.
Villa also spoke with advisors of Wigan boss Roberto Martinez, who rejected the chance to replace Houllier 12 months ago, but Latics boss Dave Whelan claimed Martinez, who held talks with Liverpool, was staying put.
Source: PA
Source: PA