Palace made the breakthrough in the fourth minute of the contest through Ismaila Sarr, with the attacker meeting a through pass from Jean-Philippe Mateta before holding off the challenge of Ian Maatsen and finishing past goalkeeper Emiliano Martinez to spark jubilant scenes in the away end.
Villa were a big threat, though, and the leveller came in the 36th minute of the contest through Ollie Watkins, with the England international making the most of a pass from John McGinn.
Dean Henderson came out to confront Watkins, but the striker rounded the Palace goalkeeper before sliding the ball into the back of the net to register an equaliser for Unai Emery's side late in the first period.
Leon Bailey then struck the crossbar for Villa, which was the start of an incredible end to the first period.
Villa midfielder Youri Tielemans saw his penalty saved by Henderson after Will Hughes had failed Bailey inside the box, and Palace then went up the other end and scored through Justin Devenny, with the midfielder sweeping the ball home after another counter-attack had ended with Sarr finding the 21-year-old.
The home side managed to level the scores in the 77th minute of the clash, with Ross Barkley heading a corner from Tielemans into the back of the net, and it finished all square at Villa Park, with Villa managing to avoid a fifth straight loss in all competitions, but it will still go down as a disappointing result for the Champions League outfit.
Villa have moved into seventh spot in the Premier League table, level on points with sixth-placed Nottingham Forest, while Palace remain 18th, now on eight points from their opening 12 matches of the campaign.
ASTON VILLA VS. CRYSTAL PALACE HIGHLIGHTS
4th min: Aston Villa 0-1 Crystal Palace (Ismaila Sarr)
Palace take a shock lead in the fourth minute of the clash at Villa Park, and it is a brilliant goal from Sarr, who races onto a through pass from Mateta before finding the back of the net with a composed finish.
36th min: Aston Villa 1-1 Crystal Palace (Ollie Watkins)
Villa level the scores in the 36th minute of the contest, and the goal comes from Watkins, who breaks onto a pass from McGinn before rounding the Palace goalkeeper and rolling the ball into the back of the net.
44th min: Youri Tielemans (Aston Villa) missed penalty
Villa have the chance to take the lead in the 44th minute of the contest from the penalty spot after Bailey goes down inside the box under a challenge from Hughes, but Tielemans's effort is kept out by Henderson.
45th min+1: Aston Villa 1-2 Crystal Palace (Justin Devenny)
Incredible scenes here; Villa miss a penalty and then Palace go up the other end and score, with Devenny finding the bottom corner with an excellent finish after Sarr had delivered a cross into the penalty box.
77th min: Aston Villa 2-2 Crystal Palace (Ross Barkley)
Villa level the scores in the 77th minute of the contest, with Barkley heading a corner from Tielemans into the back of the net. Can the home side find a winner in the final exchanges of this contest?
MAN OF THE MATCH - ISMAILA SARR
Sarr sent Palace ahead in the early stages of the contest before turning provider for Devenny late in the first period, with the attacker's dangerous cross being converted into the back of the net.
The 26-year-old was such a huge threat to Villa with his pace on the counter-attack, and he had four shots during a brilliant performance, with Sarr fully deserving of the man-of-the-match award on Saturday afternoon.
ASTON VILLA VS. CRYSTAL PALACE MATCH STATS
Possession: Aston Villa 70%-30% Crystal Palace
Shots: Aston Villa 17-13 Crystal Palace
Shots on target: Aston Villa 5-6 Crystal Palace
Corners: Aston Villa 10-1 Crystal Palace
Fouls: Aston Villa 10-11 Crystal Palace
BEST STAT
WHAT NEXT?
Villa will be looking to post their first win since October 22 when they continue their Champions League campaign against Juventus on Wednesday night, before heading to high-flying Chelsea in the Premier League next weekend.
Palace, meanwhile, will be looking to build on what was a positive result in England's top flight when they return to Selhurst Park next Saturday, with Newcastle United the visitors to Oliver Glasner's side.