Luis Diaz’s sensational strike salvaged a point for the Reds after Wilfried Zaha opened the scoring for Palace in the first half.
Just two games into the season, Liverpool trail leaders Manchester City by four points with their new £63 million (B2.6 billion) striker largely to blame.
Nunez had scored in his first two appearances against City in the Community Shield and in rescuing a point as a substitute in a 2-2 draw away at Fulham last weekend.
But the Uruguayan’s headbutt on Joachim Andersen just before the hour mark left his side with an uphill challenge in the match and increasing ground to make up already in the title race.
“He knows that now himself,” said Jurgen Klopp when asked if Nunez had let his teammates down.
“Of course it was a red card. He was provoked but that is not how he should behave.”
Liverpool partly dug themselves out of a hole thanks to Diaz’s brilliant equaliser and Klopp said he was bursting with pride at how the 10 men reacted after a difficult week caused by an early-season injury crisis.
Roberto Firmino, Thiago Alcantara, Diogo Jota, Joel Matip and Ibrahima Konate were all sidelined, while Joe Gomez was only fit enough for an appearance off the bench.
Nat Phillips was forced to make his first Premier League appearance since the final day of the 2020/21 season at centre-back, while Klopp had no options to refresh his attack late in the game.
‘Witch in the building’
“The real feeling I have at the moment is pride,” added Klopp. “With the week’s preparation we had it was unbelievable. It was like a witch was in the building, every time someone else had a problem.”
Klopp had lambasted his side for a slow start that cost them two points at newly-promoted Fulham on the opening weekend of the season.
This time Liverpool came flying out the traps, but failed to turn their early dominance into goals.
James Milner, Nunez and Mohamed Salah were all guilty of failing to hit the target with big chances, while Harvey Elliott’s goalbound shot was blocked by Andersen.
Palace had barely been in the opposition’s half for the first half-hour, but Patrick Vieira’s gameplan worked to perfection for the opening goal.
Eberechi Eze wriggled free from the Liverpool press to play the perfect ball in behind for Zaha to break the offside trap and curl beyond Alisson Becker into the far corner.
Liverpool should still have been level before half-time as Nunez mishit a glorious chance and the ball floated back off the post.
His nightmare home debut was complete on 57 minutes when he lashed out at Andersen with a headbutt that will also see him miss upcoming games against Manchester United, Bournemouth and Newcastle.
Despite their numerical disadvantage, Liverpool refused to let their proud home unbeaten record either side of the behind-closed-doors era of coronavirus restrictions go.
Diaz led the fightback with a terrific individual run and arrowed finish into the far corner to level on 61 minutes.
The red card did not deter Palace from their plan to sit deep and counter-attack, and it nearly worked.
Zaha should have won the game when he hit the post with the goal gaping from Cheick Doucoure’s cross 12 minutes from time.