Sadio Mane earned Liverpool their first Premier League win of the season as Crystal Palace suffered a second straight defeat despite an improved display at Anfield.
Jurgen Klopp's side were heading for a frustrating draw after wasting a string of chances when Mane prodded past Wayne Hennessey for his second league goal of the campaign.
Palace, beaten 3-0 at home to Huddersfield in their opening match, should have scored when the game was goalless.
Christian Benteke was unmarked and eight yards out but blazed over the bar from Ruben Loftus-Cheek's pass.
This was far from vintage Liverpool who, for once, struggled in front of goal despite peppering the Palace defence with 16 shots - half of them on target.
Joel Matip headed a great chance wide, while Mane missed a glorious opportunity before he scored - both chances created by debutant Andrew Robertson.
The full-back impressed going forward while helping Liverpool produce a much-needed clean sheet after the defensive horror show in the 3-3 draw at Watford last week.
Klopp made five changes to the team that started the Champions League play-off win at Hoffenheim in midweek and surprisingly left summer signing Mohamed Salah on the bench.
Daniel Sturridge produced flashes of his best form in Salah's place before the £34m signing came on to help his team - who were much better in the second half - to a hard-fought victory.
Mane's goal punctured the frustration which was building inside Anfield.
A one-two with substitute Dominic Solanke seemed to have been dealt with by Palace, only for the ball to roll kindly for the Senegal forward to poke past Hennessey.
Benteke scored both goals in Palace's 2-1 win at Anfield in April and the former Liverpool striker will surely still be wondering how he failed to find the net in this game.
Matip did a fine marking job on Benteke throughout. However, Benteke was well placed to give Palace a 57th-minute lead after excellent work by Loftus-Cheek but somehow hammered the ball over the bar. It was a huge let-off for Liverpool.
Nevertheless, this was a much-improved performance by Frank de Boer's team after their Huddersfield defeat, and they were well organised and solid at the back in the first half.
They also lost their opening two games of last season before bouncing back to finish seven points above the relegation zone in 14th spot.
Man of the match - Andrew Robertson (Liverpool).
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