The home side conceded an injury-time equaliser in their 3-3 draw at Watford in their opening match and had looked likely to stumble again on Saturday until the Senegal star pounced with 17 minutes left to play.
Frank de Boer endured a 3-0 home loss to Huddersfield Town last weekend in his first match in charge at Palace and, although his side produced a far more organised display on Merseyside, they were undone by another defensive error, this time from Luka Milivojevic.
Jurgen Klopp chose to start without Mohamed Salah, Emre Can and Dejan Lovren, with Liverpool facing the second leg of their Champions League play-off against Hoffenheim on Wednesday, and his team looked well short of their electric best in attack.
Joel Matip missed the best chance of the first half from a cross from debutant Andrew Robertson and Liverpool could have found themselves behind had former Reds striker Christian Benteke done better with a clear opening after the break.
Frustration around the stands was beginning to build until Mane at last showed a clinical touch to secure an important home win ahead of a difficult run of matches, which includes back-to-back meetings with Arsenal and Manchester City.
Oh Mané Mané! pic.twitter.com/iuIWoSbC7x— Liverpool FC (@LFC) August 19, 2017
James Milner, restored to the line-up on the left of midfield, tested Wayne Hennessey early on before Matip spurned a glorious chance to break the deadlock, somehow heading the ball wide of the right-hand post after the Palace goalkeeper had missed Robertson's cross.
Daniel Sturridge was tasked with leading the Liverpool attack but his only half-chance of a quiet first half hour was a shot from 20 yards out that he blazed over the crossbar.
The half at least ended in exciting fashion, with Hennessey making a tremendous save to deny Roberto Firmino from 20 yards and Simon Mignolet beating away Jason Puncheon's strike as Palace broke with speed.
Milner watched a free-kick fly narrowly over and Hennessey was at full stretch to keep Firmino's volley from creeping into the bottom corner, as Liverpool started the second half with some real attacking intent.
Benteke wasted a golden chance to score his sixth goal in four visits to Anfield as an opposition player, the Belgian skying his first-time shot from barely 10 yards out after some brilliant work from Ruben Loftus-Cheek.
Klopp introduced Salah from the bench for his Anfield debut and the Egypt winger's first contribution was to wreak havoc with a driving run, but Mane and Georginio Wijnaldum failed to capitalise.
However, with 73 minutes gone, Mane was gifted a chance he could not pass up. Milivojevic's poor control on the edge of the box saw the ball run loose, and when Dominic Solanke blocked an attempted clearance, Mane had the simple task of prodding the ball low past Hennessey.
5 - Sadio Mane has had a hand in five goals in six Premier League games vs Crystal Palace (four goals, one assist). Thorn.— OptaJoe (@OptaJoe) August 19, 2017
Liverpool's lead remained unthreatened in the closing minutes, allowing Klopp to celebrate a victory before attentions turn to the visit of Hoffenheim next week, where the Reds lead 2-1 from the first leg.
Key Opta stats:
- Frank De Boer is only the second Dutch manager to lose his first two Premier League games after Martin Jol (lost his opening three in the competition).
- Liverpool have kept five clean sheets in their last six Premier League games, as many as in the previous 21.
- Indeed, the Reds kept a clean sheet against Crystal Palace for the first time in 14 competitive matches (since January 2003).
- The Eagles have lost six of their last seven games in the Premier League (W1), failing to score in each defeat.
- Sadio Mane has netted six goals in his last eight Premier League games.
- Mane has had a hand in five goals in six Premier League games against Crystal Palace in the top flight (four goals, one assist).
- Andrew Robertson, who made his Premier League debut for the Reds, created more chances (3) and had more touches of the ball than any other Liverpool player (134).
- Liverpool have lost just once in their last 14 Premier League contests (W9 D4), with that solitary loss coming in this fixture last season.