David Moyes claimed his first win back at Goodison Park since leaving Everton, with Angelo Ogbonna impressing for West Ham.
A reunion with Jose Mourinho at Tottenham or Carlo Ancelotti at Everton have been mooted for Juventus midfielder Sami Khedira.
Edinson Cavani inspired Manchester United to a 2-0 win over Everton in the EFL Cup quarter-final, having clashed with Yerry Mina.
Manchester United are back in another semi-final after Edinson Cavani and Anthony Martial scored late goals at Everton.
Liverpool put seven unanswered goals past Crystal Palace and Arsenal lost again - this time to Everton - in the Premier League on Saturday.
Arsenal could be as low as 16th in the Premier League by the end of the weekend, and Mikel Arteta conceded his team are in a "big fight".
Everton made it three wins from three Premier League games to pile the pressure on Arsenal boss Mikel Arteta, and lift themselves to second.
Everton have made progress under Carlo Ancelotti, who has big hopes for the club as he nears a year in charge.
Kai Havertz failed to deliver against Everton, but Frank Lampard has no concerns over Chelsea's big-money signing.
The Manchester clubs showed little evidence of being title challengers in the derby, while Frank Lampard lost to one of his former bosses.