Palace was heading into the quarterfinals until Stiliyan Petrov headed in a late equaliser for the League Cup finalist.
Premier League rivals Tottenham and Bolton will also have to play their fifth-round match again after drawing 1-1. The winner will play Fulham, which advanced with a 4-0 victory over fourth-tier Notts County.
The last-eight draw also pitted holder Chelsea against either Manchester City or Stoke and Reading or West Brom against Palace or Villa.
The Premier League's bottom club, Portsmouth, will host Birmingham - unless Pompey goes out of business before the weekend of March 6-7 as it battles against a High Court liquidation order.
Palace is also enduring financial problems and the League Championship club sought bankruptcy protection last month as it tries to clear £30 million worth of debts.
Palace led twice, through Johnny Ertl's header and Darren Ambrose's free kick, but Villa hit back courtesy of headers from James Collins and Petrov - the latter coming in the 87th minute.
But Palace manager Neil Warnock claimed that the officials incorrectly awarded the corner that led to Petrov's equalizer since the ball had hit Villa striker Nathan Delfouneso.
"I am absolutely devastated," Warnock said. "Their second goal was a disgrace. We have a Premier League referee and we got (assistant) Mr. (Trevor) Massey laughing his head off all afternoon with the flag.
"If you look at it, the goalie saves it and it comes off Delfouneso and neither of them wanted to make a decision. They (the officials) wait and wait and wait to see whether they should give it. It's a disgrace, he should be banned for weeks should Mr. Massey. All he can do is smile."
Notts County, which faces a winding-up order like Portsmouth, was the lowest ranking team left in the competition - and the gulf showed at Fulham.
Simon Davies and Bobby Zamora scored to put Fulham in control at halftime and further strikes from Damien Duff and Stefano Okaka completed the rout.
"We'll be highly motivated and nervous about the quarterfinal because we badly want a good result," Fulham manager Roy Hodgson said. "It would be another major step forward with the progression at the club."
At the Reebok Stadium, goalkeeper Jussi Jaaskelainen kept Bolton's FA Cup hopes alive by correctly diving to his right to block Tom Huddlestone's 71st-minute penalty after a handball by Sam Ricketts.
"We have not been too lucky with penalties," Tottenham assistant manager Joe Jordan said. "Normally when you see Tom take them in training he does it with power - today he tried to place it. But I think when a penalty is saved you have to give credit to the goalkeeper."
Jermain Defoe, who missed a spot-kick in December against Everton and another against Leeds last month, had scored for Spurs just 10 minutes earlier. That goal cancelled out Kevin Davies' first-half strike for Bolton, which is next from bottom in the Premier League.
