It was Patrick Vieira's side who edged themselves into the lead when Eberechi Eze's perfectly executed pass found the run of Wilfried Zaha, and his finish past Alisson was equally impressive. The Reds applied pressure after the restart, with Darwin Nunez's effort striking the post, but it was the Eagles who held the lead going into the break.
A chance for Nunez after half-time saw him scuff his attempt with his left foot after Mohamed Salah played him through -- and that turned out to be Nunez's final chance of the game after he was dismissed for violent conduct against Joachim Andersen in the 57th minute.
A sensational solo effort from Luis Diaz got Liverpool back into the game shortly after as Jurgen Klopp prepared a triple substitution, and while the Reds looked to maintain control of the match, they were left frustrated and unable to fashion a significant chance that could threaten Vicente Guaita's goal.
The Reds ensured control of the match for the majority of the game and created plenty of chances, which should have been enough to take all three points.
Liverpool also didn't look to lose momentum after going down to 10 men, with Diaz in particular stepping up to the challenge.
Liverpool's high line was caught out at times, and that might be something that manager Klopp will have to alter in games where Nat Phillips is starting.
The Reds struggled to deal with the pace of Zaha who timed his runs excellently on the night. A red card for Nunez will also see him miss time in a Premier League home debut to forget.
Manager rating (out of 10; 10 = best)
Jurgen Klopp, 7 -- The Liverpool manager's system helped create a number of chances that his team simply didn't take. Klopp also reacted well to going a man down, making three changes that helped ensure control for the remainder of the match.
Player ratings (players introduced after 70 minutes get no rating)
GK Alisson, 6 -- There wasn't much that Alisson could have done about a stellar finish from Zaha. Outside of that, he was largely a spectator amid Liverpool's dominance of possession.
DF Virgil van Dijk, 6 -- The defender should have done more on Crystal Palace's goal to get tighter to Zaha, but he was otherwise faultless in his defensive duties. Van Dijk's passing was also noteworthy, with his switches helping get the ball to Salah.
DF Nat Phillips, 5 -- It was a difficult night for Phillips who made his first start for Liverpool in over a year. He was caught out by the pace of Zaha for Crystal Palace's opener, and Vieira's side continued to look to find space in behind him.
DF Trent Alexander-Arnold, 7 -- Alexander-Arnold was one of Liverpool's most creative players on the day, with his delivery from range helping create a number of chances.
MF Fabinho, 7 -- The Brazil international looked to be in impressive form and regularly broke up the play in the middle of the park before quickly moving the ball into Liverpool's more creative players.
MF James Milner, 6 -- Milner was close to scoring in the opening minutes with a left-footed strike that was sent way over the bar. He kept things simple for the majority of the game, with most of his work conducted from a deeper role.
MF Harvey Elliott, 7 -- The 19-year-old ran tirelessly in the middle of the pitch and was at the heart of some promising chances. His well picked-out pass to Nunez on the cusp of half-time should have been converted.
FW Luis Diaz, 8 -- Liverpool's ticket back into the game. Diaz's goal was simply sensational, having made his way past multiple defenders before rifling the ball past Vicente Guaita from outside of the box.
FW Darwin Nunez, 3 -- The Uruguay international's finishing should have been much better on the night, having missed a couple of glowing opportunities, but his game was defined by a foolish red card for violent conduct.
FW Mohamed Salah, 7 -- Salah always looked a threat when on the ball, and he was close to picking up an assist with a delicate through ball to Nunez, who produced a tame left-footed effort that was blocked.
Joe Gomez (Phillips, 63"), 7 -- Gomez won the majority of his duels and looked composed in possession, quickly transitioning the play into midfield.
Kostas Tsimikas (Robertson, 63"), 5 -- The Greece international was brought on to stretch the play down the left-flank, and while he got into some promising positions, his crossing should have been more accurate at times.
Jordan Henderson (Milner, 63"), 6 -- Henderson looked impressive from a deep role and picked out a number of positive passes into forward areas.
Fabio Carvalho (Elliott, 79"), N/R -- A home debut for Carvalho in which he had a volley that came close to giving the Reds the lead.