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Salah and Mane now have 21 and 20 goals each this season -- making them the top two scorers in the Premier League.
City have 89 points, one more than rivals Liverpool, with both teams having three games to play - and none of them against top six opponents.
After Tottenham Hotspur, Manchester United and Arsenal all lost at the weekend, Chelsea had an opportunity to tighten their grip on the Premier League's top four places.
The victory was an important one for Tottenham, who face a daunting trip to league leaders Manchester City in the Champions League on Wednesday.
Huddersfield Town were relegated while Manchester United celebrated the full-time managerial appointment of Ole Gunnar Solskjaer with a 2-1 win over Watford.
It was United's first league defeat since December 16.
Liverpool move to 73 points, one behind City, who beat Watford 3-1 on Saturday.
Tottenham Hotspur's challenge faltered again with a 2-1 loss at Southampton.
The Premier League battle appears a two-horse race after leaders Liverpool and Manchester City both won on Wednesday while Tottenham fell to a second straight defeat.
The win moves Juergen Klopp's side three points above second-placed Manchester City.