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The Reds went behind to a Jetro Willems strike in the seventh minute at Anfield, but a brace from Mane and a third from Mohamed Salah ensured the three points.
Firmino’s relentless pressing and excellent technique created a number of chances for the hosts, including one apiece for Trent Alexander-Arnold and Andy Robertson, who were unable to convert at the near post.
He had already laid on Mane’s second of the afternoon, with the Senegalese benefiting from a harsh bounce off Martin Dubravka to add to his brilliant curled opener.
And with Salah sweeping through as the final quarter-hour approach, Firmino played a sumptuous flick on the one-two for the Egyptian to fire home with his left.
It was a result that secures top spot after five games in the league, with Liverpool taking 15 points from a possible 15, remaining ahead of Man City.
Their next outing comes in the Champions League, with a trip to Napoli to come on Tuesday, and Klopp faces a tight turnaround as the Reds bid to defend the European Cup.
Needing to bring Firmino on so early into a game in which he had been rested will not be considered ideal, but for now the manager will be delighted his No. 9 has maintained his world-class start to the season.