Jurgen Klopp's side came into the match after starting the Premier League season with a disappointing 2-2 draw against Fulham and, despite dominating the opening half hour, fell behind when Wilfried Zaha beat Alisson with a fine finish following a quick-hitting counter from Palace.
The visitors fought off a late flurry of chances from Liverpool -- who took 17 shots in the opening 45 minutes -- as half-time approached and took their 1-0 lead into the dressing room.
Matters got worse for the hosts as they chased a second-half equaliser when Nunez was shown a straight red card for headbutting Palace defender Joachim Andersen shortly before the hour mark, leaving his team to chase the game with just 10 players.
Liverpool got back on level terms minutes later despite going down a man with a brilliant individual effort from Luis Diaz, who picked up the ball on the left flank, beat multiple Palace defenders as he tiptoed across the top of the penalty area and unleashed an unstoppable shot past Palace keeper Vicente Guaita.
Zaha had a glorious chance to put Palace back in the lead in the 78th minute, but his shot at the back post from Cheick Oumar Doucoure's cross through the area went wide of Alisson's net and the teams were forced to settle for a draw.
"Obviously we always want to win our games and especially at home," Liverpool midfielder James Milner said after the match. "I thought we started pretty well and then we lost our way a bit in the middle. They've obviously got quality, they're always going to cause you problems on the break, which we could have defended better."
The result leaves Liverpool in 12th place on two points from their first two matches, four behind leaders Manchester City, while Palace sit in 16th with a single point.
Liverpool also saw their streak of nine straight wins in Premier League home openers snapped and failed to win either of their first two top-flight matches for the first time since 2012-13.
Palace boss Patrick Vieira was delighted to leave Anfield with a point, despite having an extra man for the last 30 minutes.
"Overall we keep our organisation our discipline -- this is a really tough place to come and try to perform," Vieira said. "It was a really good team performance and I'm really happy with the team, the way they played today."