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Hours after Arsenal's 3-0 win over Bournemouth, Pep Guardiola's men responded with a resounding victory that pulled them to within a point of the North Londoners at the top of the standings, but with a game in hand.
Liverpool remain third on 75 points, one behind second-placed Manchester City, who have played two games fewer, and two behind first-placed Arsenal, who have a game in hand and a much-superior goal difference.
Foden struck twice as Manchester City stepped up their bid for a fourth Premier League crown in a row with a ruthless 4-0 victory at Brighton & Hove Albion.
Everton's first derby victory at Goodison Park in more than 13 years prevented Liverpool from climbing level with Arsenal at the top of the league table. Juergen Klopp's men are three points behind the Gunners (77) with four games to play.
It was the sixth time Arsenal have scored five or more in a Premier League game this season without conceding and in such a tight race they now have a huge goal difference advantage which could prove crucial to the outcome.
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.