But the Reds' 1-0 victory came at a cost as goalkeeper Alisson was forced off with a suspected hamstring injury and is now set for a spell on the sidelines.
Diogo Jota's first-half finish was enough to secure Liverpool's sixth successive victory in all competitions.
Arne Slot also became the first Liverpool manager to win as many as nine of his first 10 games, with the Dutchman's only setback to date being the home defeat to Nottingham Forest.
Palace could have been dead and buried by half-time as the visitors dominated.
But the Eagles had an Eddie Nketiah strike disallowed for offside and might have had a penalty when Virgil van Dijk held Marc Guehi's arm in the box.
Fulham's Rodrigo Muniz lands on his head after a clash against Man City. (AP PHOTO)
Manchester City and Arsenal trail Liverpool by one point after respective home wins over Fulham and Southampton.
City extended their unbeaten run in the Premier League to 30 games as Mateo Kovacic scored twice in a 3-2 victory.
Former Manchester United midfielder Andreas Pereira put Fulham ahead from close range after 26 minutes.
But Kovacic equalised within six minutes through a deflected effort and struck again soon after the restart.
Jeremy Doku gave City breathing space with a fantastic strike to ensure substitute Rodrigo Muniz's late effort for Fulham was no more than a consolation.
Arsenal also came from behind to beat Southampton 3-1 after Cameron Archer had given the winless Saints a 55th-minute lead.
Kai Havertz and Gabriel Martinelli struck in 10 second-half minutes to turn things around and Bukayo Saka sealed matters two minutes from time.
Brentford beat Wolves 5-3 in a thriller after the Bees again showed their liking for early goals with Nathan Collins heading them into a second-minute lead.
Wolves quickly levelled through Matheus Cunha and, after Bryan Mbeumo's penalty made it 2-1, Jorgen Strand Larsen poked home another equaliser for the visitors.
Christian Norgaard and Ethan Pinnock gave Brentford breathing space in the first half and Fabio Carvalho added a fifth late on before Rayan Ait-Nouri's stoppage-time consolation.
West Ham thumped Ipswich 4-1 to ease the the pressure on Julen Lopetegui, who has endured an indifferent start since replacing David Moyes as manager.
Michail Antonio opened the scoring inside the first minute but winless Ipswich were quickly level through Liam Delap's fourth goal of the season.
The Hammers were not to be denied though and goals from Mohammed Kudus, Jarrod Bowen and Lucas Paqueta secured their first home win of the season.
Leicester saw off Bournemouth 1-0 at the King Power Stadium for their maiden win of the Premier League campaign.
Argentinian teenager Facundo Buonanotte was the match-winner, sweeping home James Justin's pass after 16 minutes.
Everton and Newcastle drew 0-0 in the late kick-off at Goodison Park.
Former Everton forward Anthony Gordon had the chance to score against his former club from the penalty spot in the first half after James Tarkowski dragged Sandro Tonali to the floor.
But Gordon was denied by his former team-mate and England colleague Jordan Pickford, who dived to his left to make a fine save.
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