It was far from vintage form for Liverpool who led 2-0 after a dominant first half but needed a late penalty by Fabinho to quell a stirring Palace fightback at Selhurst Park.
Third-placed Chelsea got back to winning ways with a 2-0 home victory over Tottenham Hotspur at Stamford Bridge with an exquisite strike by Hakim Ziyech setting them on their way to a first victory in five league games.
Manchester City were denied a 13th successive league win on Saturday in a 1-1 draw at Southampton and now lead Liverpool by nine points having played a game more.
Chelsea are 10 behind City but have played a game more.
Arsenal missed an opportunity to move into the top four as they drew 0-0 at home to bottom club Burnley.
Leicester City drew 1-1 at home to Brighton and Hove Albion -- Danny Welbeck's late equaliser earning ninth-placed Brighton a 12th draw of the season.
Goals by Virgil van Dijk and Alex Oxlade-Chamberlain put Liverpool in charge at Selhurst Park.
But Palace grew in belief after the break and when Odsonne Edouard tapped in after 55 minutes they looked capable of salvaging something until a controversial late penalty awarded when Diogo Jota collided with Palace keeper Vicente Guaita.
"A big three points today for us, really big," Liverpool manager Juergen Klopp said, although he played down the significance of the win in the title race.
"The distance is too far to talk about it but we don't have to. We just need to win football matches. Each Premier League game is difficult like we saw today," Klopp said.
Chelsea recorded a third win this month against Tottenham, including two in the League Cup semi-final, to get back on track after a worrying slump in league form.
They were far too good for the visitors but had to wait until Ziyech's superb curled effort two minutes after the restart to go ahead. Thiago Silva made it 2-0 shortly after and Chelsea could have won by more.
Harry Kane did have a first-half goal disallowed for Tottenham but they surrendered their unbeaten league record under Antonio Conte in rather tame fashion.
"We were working extremely hard for it and it was a tough match," said Chelsea manager Thomas Tuchel, who was celebrating one year in charge of the club. "The effort was outstanding and it was a deserved win.
"It is so hard to create chances against them but we were relentless and kept on believing."
While Tottenham have slipped to seventh with 36 points, they have games in hand over all the teams battling for fourth spot.
Arsenal edged above them on goal difference but were disappointing as they could find now way past Burnley.
"To win this match you need to have a different level of quality," said manager Mikel Arteta, whose side were booed off at the end and have endured a disappointing January.
The Premier League now takes a two-week winter break with the focus switching to the transfer window that closes on Jan 31. Action resumes on Feb 5.