The Reds, winless against Palace at Anfield since October 2013, controlled the game throughout but were made to wait for the breakthrough against a dogged Eagles side.
Joël Matip and Mané should both have done better from Andrew Robertson's crosses in the first-half, while Christian Benteke spurned a golden chance for the Eagles before the hour.
But Mané finally broke the deadlock with 17 minutes remaining as the combination of a defensive error and substitute Dominic Solanke's harassing allowed the winger to stab in from close-range.
That was enough for Liverpool to secure a first win of the 2017-18 campaign, one which sees them end a run of three consecutive defeats at home to Palace and also earn their first league clean sheet against the Londoners since December 1997.
Major Reds rotation as Sturridge returns
Jürgen Klopp made five changes from the eleven that faced Watford and Hoffenheim last week. Daniel Sturridge started up front, forcing Mohamed Salah to the bench.
Joe Gomez started his first league game since September 2015 with Trent Alexander-Arnold rested, while summer signing Robertson made his debut at left-back. James Milner and Ragnar Klavan also came into midfield and central defence respectively.
Palace - having won their previous three visits to Anfield - brought in Andros Townsend to replace star man Wilfried Zaha, who has been ruled out for four weeks with a knee injury.
Frank de Boer's only other change saw James Tomkins replace summer signing Jaïro Riedewald, who dropped out of the squad entirely. Midfielder Yohan Cabaye returned to the bench.
Matip and Mané fail to make more from Robertson's deliveries
After a slow start, centre-back Matip should have headed the home side in front after 15 minutes.
A short free-kick saw Robertson exchange passes with Milner before curling a ball into Matip, who - seemingly distracted by Wayne Hennessey's outstretched hand - got his angles badly wrong to head wide of an open goal.
Liverpool had plenty of typically fluid attacking combinations as they controlled possession and looked to break through Palace's back-line with clever build-up, but they were starved of space and goalscoring opportunities as a result.
Stung by their thrashing at home to Huddersfield Town last weekend, the Eagles' defensive shape deliberately dulled proceedings and kept the Reds from translating their dominance into an advantage.
Liverpool went close again after another of Robertson's left-sided crosses, this time from a short corner, but Mané could not adjust himself in time to get a shot away inside the six-yard box.
Palace thwart home side but Benteke wastes excellent chance to make it 1-0
Hennessey and Simon Mignolet were then both called to beat away powerful attempts inside 20 seconds of each other as Roberto Firmino and Puncheon drew saves at either end of the pitch.
But that was a rare instance of end-to-end action in an uneventful first-half short of cutting edge, with Robertson's dangerous deliveries from the left Liverpool's only real threat.
The second-half was much the same, Milner wasting a free-kick in a promising central position by curling over the crossbar after Matip had been upended on the edge of the area.
Gini Wijnaldum forced a good save from a corner with a 25-yard volley, though it lacked enough power to really trouble Hennessey who got down to the attempt comfortably.
Ex-Liverpool striker Benteke should have punished his former club's toothlessness before the hour mark and added a sixth goal in four just visits to Anfield as an opposition player.
Ruben Loftus-Cheek burst in behind and managed to shrug off Klavan's attentions to break into the box and square for Benteke, but the Belgian somehow side-footed over the bar from eight yards.
That was a huge let-off for Liverpool, who stepped up their intensity as captain Jordan Henderson forced a save with a long-range curler after Firmino had drilled an angled drive just over the crossbar.
Klopp summoned Salah for his first Anfield appearance to improve Liverpool's fortunes, taking off Sturridge.
Mané breaks the deadlock after Solanke's introduction
Mané and Wijnaldum both saw shots blocked by Patrick van Aanholt in the box on 70 minutes before Hennessey easily denied the former as the Merseyside outfit grew closer to a first goal of the afternoon.
That was Wijnaldum's final involvement before summer signing and pre-season stand-out Solanke made his Premier League debut from the bench.
And only a minute later, Solanke had helped to force the breakthrough. Luka Milivojević's poor touch on the edge of his own box - combined with pressure from Solanke - forced the ball into Mané's path and he reacted quickly to prod into the bottom corner.
That was the Senegalese winger's sixth goal in his last eight league games and forced Palace to try and open the game up late on, Sullay Kaikai coming on for just his third first-team appearance.
De Boer even threw defender Scott Dann up top alongside Benteke as Palace opted for a direct approach, but it was too little too late for the visitors.
Solanke, Robertson and Salah all went close to making sure of the win late on but, unlike at Watford a week ago, Liverpool rode out the latter stages to secure a hard-earned three points.
Palace remain winless from their first two games but were improved upon their display against Huddersfield. Their downfall was a lack of threat up top, with just one shot on target all game and Benteke wasting their best chance.
They take on Ipswich Town in the second round of the League Cup on Tuesday, while Liverpool will look to confirm their place in the Champions League group stages in the second-leg of their qualifier at home to Hoffenheim - leading 2-1 from the first-leg.