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The heavy defeat leaves Everton 16th on 27 points, two places and two points above the relegation zone.
Manchester United fanned the embers of their top-four hopes with a 2-0 win against visiting Everton.
Liverpool have 60 points with champions City on 59 and Arsenal on 58, all three teams having played 26 games.
Liverpool's patched-up side came out after the break in top gear.
Liverpool won 3-1 while Manchester City beat Everton 2-0 in an earlier match.
Liverpool, beaten by Arsenal on Sunday, lead the table on 51 points from 23 games but City are up to 49 from 22 and Arsenal on 49 from 23.
Fourth-placed Aston Villa suffered a 3-1 home defeat by Newcastle United, Crystal Palace came from behind to beat visiting Sheffield United 3-2 and struggling pair Fulham and Everton fought out a goalless draw at Craven Cottage.
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The 31-year-old Salah, who has 151 Premier League goals for Liverpool, changed his boots at halftime and the switch paid off handsomely.
Pep Guardiola's newly-crowned FIFA World Club champions had lost ground in the title race after a run of one win in six league games.