Liverpool had to wait until the 73rd minute to get the winner, the ball fortuitously falling to Mane inside the box after Luka Milivojevic's mistake on the edge.
Christian Benteke missed a clear opening for Palace earlier in the second half from 10 yards, blazing over the bar for the visitors' best opportunity.
Liverpool, who were without Philippe Coutinho again with his future still uncertain, held on for their first league win of the season, while Frank de Boer's new Palace side are without a point after two games.
Both sides were looking for their maiden Premier League victory of the campaign, and it was the hosts who had the best chance in a goalless first half.
The impressive Andrew Robertson, making his debut for Liverpool, sent a superb ball in from the left, but Joel Matip somehow headed wide from three yards after Wayne Hennessey had misjudged.
Roberto Firmino forced Hennessey into a fine stop from long range later in the half, before Palace went up the other end and saw Jason Puncheon's effort from an angle beaten away by Simon Mignolet.
Hennessey was at his best again in the second half, saving from Gini Wijnaldum and Jordan Henderson's long-range effort, but Palace should have been ahead on 57 minutes through former Liverpool striker Benteke, who struck over the bar after Ruben Loftus-Cheek's fine work on the right byline.
Liverpool piled on the pressure and got their reward with 17 minutes remaining as Milivojevic bolted under the pressure of sub Dominic Solanke, allowing the ball to fall luckily into the path of Mane, who made no mistake low into the net.
Solanke, Robertson and Mohamed Salah could have doubled Liverpool's lead late on, but all three were denied by Hennessey as Palace stayed in the game until the final moments.
The striker was scoring his first Premier League goals since April 2014 (1232 days ago).
Amazingly, all 39 of his goals in the English top-flight have come inside the box. Indeed, only Tim Cahill (56) and John Terry (41) and have scored more Premier League goals without ever netting one from outside the box than Hernandez.
Despite playing the majority of the game with 10-men, the Hammers made the most of their limited territory with Hernandez finishing the only two chances that came his way - both from inside the six-yard box. His all-round game was also of top quality, including when mucking in as a makeshift left-winger.
It's good to have him back in the Premier League.
Manchester United continued their perfect start to the Premier League season by crushing Swansea City 4-0 with three late goals in four minutes at the Liberty Stadium on Saturday.The scoreline matched their opening-day destruction of West Ham United after Romelu Lukaku, Paul Pogba and substitute Anthony Martial scored between the 80th and 84th minutes to add to Eric Bailly's first-half opener, his first goal for the club.Lukaku began the late rout with a neat finish on 80 minutes for his third goal in two games before Pogba surged forward to dink the ball past Lukasz Fabianski and Martial completed the destruction.Two of the goals came from assists by Henrikh Mkhitaryan, who was named man of the match."It was a tough game, especially in the first half when they had five defenders," the Armenian said."In the second half there was more space as they only played with four."Despite raucous home support, Swansea seldom threatened in the absence of the departed Gylfi Sigurdsson the injured Fernando Llorente, always struggling to deal with United's power and pace and by the end Jose Mourinho's side were bursting forward almost at will."I think we are a more confident side this season," said Pogba, who gave a strong performance despite playing most of the match with a yellow card after a wild early challenge."Last year we still won two trophies but now we all know each other better and are working for each other."His words reinforced the feeling that United are a much stronger force this season, not least because of their strength on the bench.Last week they scored two late goals to finish off West Ham and on Saturday, after injecting Martial into the attack on 75 minutes, they went one better.
Tottenham Hotspur are targeting four new signings in the transfer window but need time to recruit the right players to help the club progress, manager Mauricio Pochettino said ahead of Sunday's match against champions Chelsea.Spurs, who were second in the Premier League last season, are the only English top-flight club yet to sign a player and have sold right back Kyle Walker to rivals Manchester City.But they are close to completing their first signing, having announced on Friday that they have agreed a fee for Ajax Amsterdam's 21-year-old Colombia defender Davinson Sanchez, and Pochettino remains hopeful of signing more players."I want four new players if it's possible but it's not easy, you need time," the Argentine told reporters on Friday."We expect to sign some players before the end of the transfer window... and try to improve our squad and bring new faces and energy in the team because it will be a long season."The most important thing is to try to be right in the players we sign and that the players can help us and be important for the present and the future of the club."Tottenham have been linked with Everton's unsettled England international Ross Barkley and Celta Vigo's Papa Chiekh Diop.Spurs will play their home games at Wembley this season while their new stadium at their White Hart Lane site in north London is under construction and due for completion next year.However, their record at the venue last season was underwhelming, as they lost two and drew one of their four European fixtures at the national stadium.Spurs also lost to Chelsea in the FA Cup semi-final at Wembley in April and Pochettino said his team would have no excuse for poor performances at the stadium this campaign."One of our challenges is to make Wembley our home. We need to show we are ready," he said. "Wembley is not a problem. We are the problem if we are not able to win like last season. We were poor when we played there last season."Pochettino believes that playing against the league champions on Sunday will make his team more determined."Chelsea have a very good squad. They have signed good players. It will be a really tough game, a big challenge... it's always very exciting to play in a derby," he said."It's a massive motivation for us and our fans to play against the Premier League champions."
Champions Bayern Munich eased past Bayer Leverkusen 3-1 in the Bundesliga season opener on Friday to lay down a marker as they eye a record-extending sixth straight German league title.Newcomers Niklas Suele and Corentin Tolisso put the Bavarians ahead before Robert Lewandowski added a third goal with a penalty, following the official use of the video assistant referee for the first time in the Bundesliga.Bayern were missing close to half a dozen key players through injury including keeper Manuel Neuer, Thiago Alcantara and Jerome Boateng, but it never showed as they rode roughshod over Leverkusen for more than an hour at the Allianz Arena.Admir Mehmedi pulled a goal back in the 65th minute for the visitors, who missed a handful of late chances to score again.Bayern could have taken the lead after five minutes when Frenchman Tolisso, the club's record signing at 41 million euros (37.46 million pounds), came close to scoring on his league debut but goalkeeper Bernd Leno did well to block his low drive.It took the hosts another five minutes to break the deadlock when Suele head home after fellow newcomer Sebastian Rudy floated a free kick to the far post.Midfielder Tolisso did score in the 19th minute with a close-range header from Arturo Vidal's clever cross.Tolisso then rattled the post four minutes later with Leverkusen struggling to repel the Bayern pressure.RAIN DELAYThe start of the second half was delayed for more than 20 minutes due to torrential rain, but seven minutes later the Bundesliga's first ever use of the video assistant referee handed Bayern a penalty.Poland forward Lewandowski, who had been brought down by Charles Aranguiz, converted the spot kick to grab Bayern's third goal before Mehmedi fired in from close range.Leverkusen finally managed to spark into life but were wasteful with their chances as Bayern stepped off the gas.Last season's runners-up RB Leipzig begin their title challenge at Schalke 04 on Saturday while Borussia Dortmund, who finished third, travel to VfL Wolfsburg.