It is believed the Midlands clubs have paid around £1 million in compensation.
Paul Lambert leaves Norwich City to manage Aston Villa
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 today 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 £1 million (Dh5.6m) 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.
"Obviously last season wasn't the best for a team as big as us. We should have been way higher up the league and I think (the board) see Lambert as an up-and-coming manager that is doing really well, so it is going to be a massive lift for everyone, the fans and the players as well.
"It is a fresh start for everyone again and hopefully in pre-season everyone is going to have a chance to prove their worth to the club."
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.
It is believed Lambert will be given money to spend after McLeish spent last season reducing what he described as an "astronomical" wage bill.
Lambert will also be able to use the money he generates from the sale of current Villa players.
Despite having rejected Lambert's resignation, Norwich were already preparing for his departure and had begun their search for a new manager after placing the Scot on gardening leave.
Former City defenders Malky Mackay, who led Cardiff to the Carling Cup final and the npower Championship play-offs last season, and Steve Bruce, sacked by Sunderland in November, are thought to be among the front-runners for the job along with Celtic's Neil Lennon.
Chief executive David McNally denied an approach had been made to any of the clubs concerned for permission to speak to their managers, but confirmed the search was under way even before today's announcement.
"We are looking for a new manager, that is likely what we concentrate on in the next few weeks," McNally told Sky Sports News yesterday.
"We were in a situation where we were approached by another club to speak to Paul Lambert, we didn't grant them permission and we were very clear we wanted to keep our manager who has been incredibly successful.
"When it was indicated by our manager that he would like the opportunity to speak to this particular club the dynamics changed, and then it was about talking to them about compensation, agreeing compensation and sorting out a time frame."
He added: "Paul Lambert has been fantastic for this football club for three years and he will quite rightly take his place in Norwich City history and we will always be very grateful to Paul for his achievements here."