The announcement came on the day the club said it had begun working with former Football Association executive Adrian Bevington.
A reshuffle of the boardroom has seen Birmingham businessman Steven Hollis become the club’s new chairman, former Governor of the Bank of England Lord King and ex-Football Association chairman David Bernstein become directors, and former manager Brian Little join a more football-orientated board.
Chief executive Tom Fox and sporting director Hendrik Almstadt left the club earlier this month.
Owner Randy Lerner put Villa up for sale last May, but is yet to find a buyer.
Fans have been upset at the lack of investment and the decision-making at the club.
Former Lyon boss Garde was said to be considering his future at the club after being unable to make any signings during the January transfer window.
Garde had become increasingly dispirited by performances and said he felt “humiliated” after a 6-0 defeat by Liverpool last month. It was the first time the Midlands club had let in six at home in a league game since October 1983.
His only victories came against Norwich City and Crystal Palace in the league, and League Two Wycombe Wanderers in the FA Cup.
He recently refused to comment on reports Lyon president Jean-Michel Aulas is keen for him to return to the Ligue 1 club next season.
Villa’s next Premier League match is against Chelsea at home on 2 April.
Both sides are fed up of each other – analysis
BBC Sport’s Pat Murphy said there was “mutual antipathy”, with “both sides fed up of each other” and discussions continuing over a compensation payment for Garde, who was manager for 147 days.
“Garde was promised players when he signed,” Murphy told BBC Radio 5 live. “Lerner turned the tap off and Garde felt badly let down.
“He’s looked more and more downcast. His body language has been graphic.”
Asked about the timing of the decision, announced during Tuesday’s England v Netherlands international, Murphy said: “A good day to bury bad news? I honestly don’t think so.
“Negotiations have been tortuous and labyrinthine and he would have had to face the media on Thursday with a match on Saturday.
“I think it’s just the usual cock-up – yet another farcical period in Villa’s recent history.
“Garde didn’t think much of the players and the feeling was mutual.
“I don’t think Villa could go any lower. But the players who got them into this situation are still standing – they should be looking over their shoulders.”
Who are the contenders to replace Garde?
Murphy says Villa approached former Everton, Manchester United and Real Sociedad manager David Moyes last year, but the Scot was “unimpressed”.
However, Murphy says Moyes – who was sacked by the La Liga club in November – would now “like talks”.
Other candidates would include managers who “know the Championship” added Murphy, including Burnley’s Sean Dyche, Hull’s Steve Bruce, Ipswich boss Mick McCarthy and former Leicester manager Nigel Pearson.