Villa have pressed ahead with the appointment of the 52-year-old despite a backlash from supporters and a row with Blues.
McLeish told the club's website: "I am honoured to have this opportunity to manage a club with such a fantastic history as Aston Villa's.
"The heritage, the history of success and the tradition of Aston Villa are compelling and irresistible. The challenge for me is to try to add my own chapter.
"Since first becoming a manager at Motherwell, my desire to succeed and my energy and drive have never waned.
"My objective is to impart that drive and will to win to the players and fans of Aston Villa.
"I know that some of our fans have voiced concerns and I can understand why.
"It will be up to me to convince you that I am the right man to drive the club forward and I intend to give absolutely everything to prove that I can be a success at the club."
Villa have not yet confirmed the length of McLeish's contract but have said the Scot will commence duties immediately, underlining their confidence there are no lingering issues concerning Birmingham.
Blues were angered when it became apparent McLeish was a contender for the Villa vacancy and accused their rivals of tapping up the former Rangers and Scotland boss.
They also demanded ?5.4million in compensation to cover the remaining two years of his contract but threats to report Villa to the Premier League or take out an injunction to block the move were not carried out.
Villa, who switched their attention to McLeish after being snubbed by Wigan's Roberto Martinez last week, understood McLeish to be a free agent.
McLeish becomes Villa's third full-time manager in nine months and the club need him to restore stability after a difficult season.
Martin O'Neill resigned just five days before the 2010-11 campaign began and his successor Gerard Houllier made a number of PR gaffes before leaving after a health scare.
Villa were also not deterred by the reaction of their own fans, who have been incensed at the prospect of McLeish moving across the city.
Their objections culminated with a protest outside Villa Park on Wednesday night.
Villa chief executive Paul Faulkner said: "We believe we have appointed the right man for the job.
"Unquestionably, Alex meets the criteria we set out at the beginning of our search which was based on proven Premier League experience, leadership, a hard-working ethic and, most importantly, a shared vision for Aston Villa.
"Alex is someone with whom we will work closely and work well together.
"With our strong squad combining the necessary virtues of experience and the exciting potential of our young players, our objective is to compete as strongly as we possibly can.
"Alex's vast experience and proven abilities demonstrate clearly that he is a strong leader and an ambitious man and we are looking forward very much to the exciting new season about to start."
The appointment ends a long search that has continually wrong-footed bookmakers.
Mark Hughes was installed as the early favourite after leaving Fulham but was never a serious contender.
Steve McClaren, David Moyes, Rafael Benitez and Martinez were all heavily backed at different times but fell by the wayside.
McLeish initially seemed an unlikely candidate but speculation mounted after he tendered his resignation on Sunday.
McLeish had been in charge at Birmingham since 2007 and won the Carling Cup last season despite seeing the campaign end in the disappointment of relegation.
Source: Team Talk
Source: Team Talk