The hosts started the game slowly to give Patrick Vieira's side cause for optimism; Alisson was forced to tip Wilfried Zaha's effort onto the post in the opening five minutes before a flurry of dangerous balls into the box had to be cleared by the defence.
Liverpool then found some control in the game, forcing a number of good stops from Vicente Guaita, but the Palace goalkeeper couldn't do much more as he pushed Mohamed Salah's powerful header into the path of Sadio Mane, who finished calmly.
It was a similar story in the second half as Jurgen Klopp's side had to cope with further dangerous moments of play. Odsonne Edouard's touch was heavy enough to allow Alisson to clear the danger before Mo Salah struck at the back post from a corner.
A spectacular strike from Naby Keita sealed what was a hard-fought victory for the Reds against a Palace side whose performance was much better than the scoreline suggested on the day.
Liverpool regained control of the match after weathering an initial spell of pressure and showed resistance against a dangerous Palace side. The Reds intricate play on the wings generated plenty of chances and that was significant in swinging the pendulum and taking control of the momentum at vital moments.
The Reds could have been 1-0 down on another day after a slow start. Some fouls were careless and provided a number of free-kicks in dangerous areas for Palace. But the biggest negative of the day will be the injury to Thiago to Klopp's side having already Harvey Elliott last week at Leeds.
Manager rating out of 10
Jurgen Klopp 7 - Klopp's midfield selection was positive and helped support a back four that had not played together before. The Liverpool manager's game plan looked to work as the home side created plenty of chances.
Player ratings (1-10; 10 = best, players introduced after 70 minutes get no rating)
GK Alisson 8 -- The Brazilian was called into action early with a palm onto the post and that was the start of a flawless performance overall. Clutch.
DF James Milner 8 -- No full-back looks forward to taking on Wilfred Zaha but the versatile 35-year-old stood up well to the task. Milner also delivered a number of good crosses into the box, proving to be a reliable cover option for Trent Alexander-Arnold.
DF Ibrahima Konate 6 -- Grew into the game and performed well when called upon in a 1 vs. 1 battle with Christian Benteke, but the former RB Leipzig defender's debut wasn't without a couple of shaky moments.
DF Virgil van Dijk 7 -- A typical performance from the centre-back. Van Dijk looked a calm head amidst early pressure and helped Konate settle into the game on his debut.
DF Konstantinos Tsimikas 7 -- There are elements that the left-back will look to improve on defensively, but it was a much different performance at the other end of the pitch. Tsimikas linked well with Sadio Mane and troubled Crystal Palace with some astute movement off the ball.
MF Fabinho 7 -- Anchoring the midfield helped ease some moments of Palace pressure, but the Liverpool No. 6 was also impressive going forward, progressing the ball into the Reds' creative players on a number of occasions.
MF Jordan Henderson 8 -- The Liverpool captain was involved regularly in an overall strong midfield performance. Rotations on the right-flank helped develop some promising spells of link-up with Salah, and there was plenty of quality with deliveries into the box.
MF Thiago 7 -- The Spain international complimented the Liverpool midfield in a game where the 30-year-old got forward regularly. Unlucky not to score with a header that was stopped well by Guaita. A non-contact injury just after the hour mark forced Thiago to be substituted.
FW Mohamed Salah 8 --. Salah's powerful header meant Guaita could only parry into the quick-thinking Sadio Mane, who put Liverpool ahead. And the star man was involved in the second goal as he struck home at the back post to put the Reds 2-0 up. That goal capped off an exceptional performance, particularly with the winger's work rate that helped relieve spells of pressure.
FW Sadio Mane 8 -- The latest addition to the Liverpool 100 club. Mane's reaction finish from Guaita's fingertip save helped put the Reds into the lead just before half-time.
FW Diogo Jota 6 -- The Liverpool frontman was presented with a golden chance in the first half and blazed over from inside the six-yard box. Despite some good runs in behind and astute ball control, decision-making could have been better at times -- notably when the Portugal international failed to pick out a pass with both Van Dijk and Mane to his left.
Naby Keita (replaced Thiago, 62") 6 -- Keita's spectacular finish from outside of the box put the game to bed for Liverpool. One of the best goals of the season so far from Jurgen Klopp's side.
Curtis Jones (replaced Jota, 76") N/R -- The academy graduate helped settle the game down after the Reds scored their second goal of the game.
Divock Origi (replaced Henderson, 88") N/R -- On to help see out the remaining minutes of the game, providing height up front for an out-ball.