Press Association Sport understands Lambert has signed a three-year contract with the midlands club. It is believed Villa have not paid any compensation after Lambert resigned as Canaries boss on Thursday having been refused permission to speak to Villa.
Jonathan Fear, of the leading Villa fans website 'Vital Villa', said: "I think it is a smashing appointment. He was the number one target and they were prepared to wait for him. It is a good, solid appointment and he can have a look from top to bottom at what is needed at Villa."
He continued: "He has worked wonders at every club he has been at. We had a poll on our site of 14 managers and he led from start to finish as the first choice. It is the opposite of last year and we've got a fans' favourite coming in this time."
Leading German manager Ottmar Hitzfeld - now manager of Switzerland - has praised the qualities of Lambert, whom he managed when in charge of Borussia Dortmund.
Hitzfeld, who led Dortmund and Bayern Munich to Champions League success, said: "I have always followed Paul since he left Dortmund, and I congratulated him on his promotion to the Premier League with Norwich City.
"As a player, he always thought like a coach. He was a leader so it is no surprise that he has become a successful manager. Paul was a very good team player. That is one of the most important qualities as a coach and manager of a team."
Villa confirmed the appointment via a one-line statement on their official website. It read: "The board of Aston Villa are delighted to confirm that Paul Lambert has been appointed Villa manager."
Lambert fits the profile of the young and hungry manager wanted by club owner Randy Lerner. But it was chief executive Paul Faulkner who has played a leading role in securing Lambert's services and he was this week elected onto the FA Council.
Villa held talks with Molde boss Ole Gunnar Solskjaer and representatives of Wigan manager Roberto Martinez. But the club stress Lambert was always the first-choice target to replace Alex McLeish, who was sacked nearly three weeks ago.