Mane scored in the 10th minute and Firmino scored in the 45th, but a 63rd-minute gaffe by Liverpool goalkeeper Alisson cost him a fourth consecutive Premier League clean sheet when Rachid Ghezzal slotted home.
Defender Virgil van Dijk's back pass went wide under pressure, and Alisson went to play it in the left corner before being challenged by Leicester's Kelechi Iheanacho.
The keeper could not dribble away from Iheanacho, who gained possession and passed the ball ahead to Ghezzal as the midfielder scored his first Premier League goal.
Mane opened the scoring with his fourth league goal in as many games with a quick left-footed flick on a through ball from Andy Robertson, and Firmino doubled the lead just before half-time from a header following a corner by James Milner.
The assist was Milner's 80th in the Premier League, moving him into joint-seventh place with David Beckham for the most all-time, and it came in his 100th appearance for Liverpool, making him the fourth player with as many league appearances with three different clubs.
Liverpool last won their first four league games in 1990-91, two years before the Premier League was established, and have never opened a season with four consecutive clean sheets.
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