Villa parted company with the Scot, who took charge in June 2012, on Wednesday after the club dropped into the Premier League relegation zone following a 2-0 defeat at Hull City.
A return of just 12 goals and five wins from 25 league matches this season saw the pressure gradually increase on the former Norwich City boss before Villa's board decided to act.
In a statement realised via the League Managers' Association on Thursday, Lambert said: "I am extremely proud to have managed Aston Villa, a founder member of the Football League, and this sentiment will always remain with me.
"My initial remit was to conduct a massive overhaul of the playing squad, lower the overall wage structure of the playing staff and achieve this whilst keeping the club in the Barclays Premier League.
"There was also a concerted effort to purchase and develop younger players who would provide a solid footing for the football club to move forward and enhance the value of the playing squad in the future.
"When I came on board the club's owner, Randy Lerner, warned me that I was embarking on the toughest challenge of my working life and he was not wrong.
"But I have never stepped away from hard work and I put my heart and soul into the job from my first day until my last.
"I'd like to thank Randy Lerner for not only providing me with the opportunity to manage one of the most illustrious clubs in world football but also for the support and friendship he offered me throughout my time at the club.
"To the players, my sincere thanks to each and every one of them. They are a fantastic group of players and I wish them every success for the rest of their playing careers. I hope to see many of them achieve great things in the game.
"Finally, I pay tribute to the supporters who are among the most passionate I have ever encountered. They rightly hold huge expectations for their beloved football club and I sincerely hope they are rewarded with the success they deserve.
"I completely understand their frustrations and always shared their view that the football club is too big not to be competing at the top end of the table. I hope that can happen.
"You never stop learning in football management and I certainly believe the invaluable experience from my time as Villa manager will prove hugely beneficial in the next stage of my managerial career."
