There was little to choose from between the two sides with Villans stopper Emi Martínez making just one real save in the first half to deny new signing Brennan Johnson.
Adam Wharton played Johnson in behind the Villa defence but Martínez was off his line quickly to tip his shot over the crossbar.
Ollie Watkins, meanwhile, forced Dean Henderson into a super stop at the other end in their best chance of the tie.
There were more openings in the second stanza as both teams looked for he win, but Watkins could only head Matty Cash's cross against an upright with six minutes of the match remaining.
The result sees Villa remain in third place on the league standings with 43 points from 21 games - level with second-placed Manchester City and five points adrift of leaders Arsenal, who have played a game less.
Palace are down in 13th place with 28 points.
Reflecting on the dropped points, Villa boss Unai Emery told his club's website that he was happy with his team following a competitive encounter.
He said: "We were winning a lot of matches but how we competed today as well was really fantastic. We dominated the first half, had chances to score and didn’t concede a lot.
"In the second half, we started with the same idea and started dominating again. We lost 15-20 minutes when they pushed, creating some chances to score.
"In that moment we were struggling, but we reacted fantastically in the last 20 minutes and created chances, hitting the post through Ollie Watkins, a very good opportunity for Morgan and Emiliano Buendía.
"We competed fantastic. In this match we were closer to winning but we drew, and we must accept it. How we are overall in the table on day 21 is fantastic."
Palace manager Oliver Glasner was, meanwhile, proud of his players for holding Villa to a goalless draw.
He said the Palace TV: "It was a very good point today and really I give credit to the players.
"I'm very proud of every single player on the team. Because of all the circumstances... playing here against maybe the most in-form team in the Premier League.
"And then getting this well-deserved point.
"I think everybody could see and feel that in the first-half especially, we didn't have a lot of confidence.
"I think our fans were surprised that we had so little confidence. But then they helped us and pushed us, so we could get back into the game in the second-half. It was a very good performance.
Next up for Villa is a FA Cup clash with Tottenham this weekend as Palace play Macclesfield.
Tottenham captain Cristian Romero will sit out Saturday's FA Cup third-round tie against Aston Villa after being handed an additional one-match suspension by the Football Association.
Crystal Palace manager Oliver Glasner has confirmed the club would consider selling captain Marc Guehi during the January transfer window if their valuation is met, with Manchester City weighing up a move for the England defender.
Aston Villa reinforced their push up the Premier League table with a commanding 3-1 home victory against Nottingham Forest at Villa Park on Saturday.
Crystal Palace have underlined their ambition by completing the club-record signing of Brennan Johnson from Tottenham Hotspur for 35 million pounds.
