The Reds move into 8th with the win, on seven points.
Jamie Vardy scored the lone Foxes goal, as Leicester City stays on four points.
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It didn’t take long for Liverpool to give the new stand a delight, as James Milner played a pass through several Foxes to Firmino.
The Brazilian then darted across the 18 before hitting a ball across his body and past Kaspar Schmeichel. 1-0, 13′.
Liverpool was lively in attack during the first half hour, with Daniel Sturridge generating chances but not the finish.
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Mane was the beneficiary of a gorgeous bit of combo play than finished when Daniel Sturridge no-look backheeled a square ball inside the 18 for a simple-enough finish.
But Vardy, as he is wont to do, took advantage of a loose ball with clinical precision to make it 2-1 heading into halftime.
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Lallana put this one to bed with an absolute screamer in the 57th minute, taking a lay-off from Georginio Wijnaldum and hitting a ball that Schmeichel had no hope to stop.
Leicester came close to making it a one-goal game moments later, but Simon Mignolet stopped Vardy in tight.
Firmino’s second came when Mane danced around an adventuring Schmeichel and moved a slow pass into the six. The Brazilian moved into position to complete the score line.