Jürgen Klopp issued a familiar update on the Philippe Coutinho saga following Liverpool’s first home game of the season. “Nothing has changed,” he said. The same applied to his team’s performance against defensive opposition until a gift from Luka Milivojevic and a predatory finish from Sadio Mané lifted the curse of Crystal Palace here.

Liverpool’s first league win of the campaign and first home win over Palace in four years was merited. Their focus was admirable, too, given Barcelona’s persistent pursuit of Coutinho produced a third bid for the injured Brazil international, and third rejection from Liverpool, on the eve of the game. But it took that unforced error from Milivojevic and a glaring miss from Christian Benteke against his old club to ease Liverpool’s recurring frustrations on home soil.

Frank de Boer almost succeeded where many others have led in thwarting Liverpool with a defensive display, yet Palace’s difficult start continued courtesy of their own failings. “We played very concentrated for 95 minutes,” said Palace’s new, and winless, manager.

“The chances we get, you have to punish them. Christian had the biggest one. I said to the team if you keep playing with this kind of intensity and discipline the results will come. To lose 1-0 is very disappointing because we deserved more but I am convinced if we show this every week the results will come.”

Liverpool’s victory rewarded their manager’s faith in his resources. The importance of the Champions League playoff and the effect of Tuesday’s first leg in Hoffenheim was writ large on the Liverpool team-sheet. Klopp made five changes to the side that started the midweek win in Germany, shifting Mohamed Salah to the bench and resting Trent Alexander-Arnold in readiness for the return leg on Wednesday. In their place came Daniel Sturridge and Joe Gomez, though it was Andy Robertson at left-back who made the biggest impression.

Facebook Twitter Pinterest Joël Matip misses a chance as Liverpool struggled to break Crystal Palace down. Photograph: Jason Cairnduff/Action Images via Reuters

Liverpool’s £10m summer signing from Hull City adapted well on his full debut. The quality of his crossing immediately stood out and will prove a potent asset for Liverpool this season and they should have capitalised swiftly. Wayne Hennessey, the Palace goalkeeper, failed to connect with an inch-perfect Robertson delivery from the left leaving Joel Matip with a free header in front of an unguarded net. The defender headed wastefully wide.

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James Milner tested the visiting keeper, Mané miscued from another Robertson cross and Sturridge struck the wall with a free-kick but the changes had an inevitable impact on Liverpool’s rhythm. Palace also take credit for a more resilient showing than at home to Huddersfield. Palace’s formation remained the same as in the opening day 3-0 defeat but their defensive duties were executed properly here, albeit while sacrificing support to Benteke, Andros Townsend and Jason Puncheon in the opposition half.

“The football could have been much better in the first half, 100%,” said Klopp. “But I was happy with the chances we created in the second half. The first half was about making the right decision at the right moment. We showed the players at half-time where the spaces are ‘one pass and then the world is much nicer’ and they used that in the second half. The crosses from Robbo especially were good but in the first half we had nobody in the box.”

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There was more urgency to Liverpool’s performance after the restart. Georginio Wijnaldum and Jordan Henderson forced Hennessey to save at full stretch and the introduction of Salah for Sturridge restored balance to the home attack, allowing Firmino to return to a central role. But it was Benteke who had the best chance of the game before Mané’s winner. Ruben Loftus-Cheek rolled Ragnar Klavan inside the Liverpool area before cutting the ball back from the by-line. The centre-forward timed his run perfectly but the same could not be said of his finish, blazed over from seven yards with only Simon Mignolet to beat.

It was a rare opportunity Palace could not afford to squander, although their eventual punishment was self-inflicted. Patrick van Aanholt blocked from Mané and Wijnaldum in a goal-mouth scramble but just as the travelling support began to sense a shutout, and the home crowd became edgy, Palace imploded. Or, to be precise and blunt, Milivojevic imploded to cost his team.

The Serbia midfielder was under no pressure when he intercepted a cross-field ball facing his own goal. Two careless touches, however, carried the ball away from Milivojevic and into the path of Mane who seized on the invitation to sweep beyond Hennessey at close range.

“I don’t think it was a lack of concentration but he was out of control,” said De Boer. “I don’t know exactly what happened. Suddenly it was at the feet of Mané.”