Swansea City came from behind to earn a 1-1 draw against Crystal Palace in the Premier League, with Jordan Ayew scoring the leveller.
Stoke City secured a valuable three points against fellow Premier League strugglers West Brom, with Joe Allen proving key.
Sergio Aguero scored a brace as Manchester City hammered struggling Bournemouth 4-0 at the Etihad Stadium on Saturday.
Southampton's winless run extended to six Premier League games as Huddersfield Town earned a draw thanks to Laurent Depoitre's header.
Chelsea had 25 shots without success in the 0-0 draw at Everton and Antonio Conte insisted playing without a striker was not the issue.
Manchester City striker Sergio Aguero became the club's all-time leading scorer earlier this season and now has 100 Etihad Stadium goals.
Everton 0 Chelsea 0: Antonio Conte's Chelsea dominated Everton for most of the match at Goodison Park, but they were made to settle for a 0-0 draw.
Antonio Conte's Chelsea dominated Everton for most of the match at Goodison Park, but they were made to settle for a 0-0 draw.
Chelsea must pay great attention to Everton star Wayne Rooney, head coach Antonio Conte warned.
Jose Mourinho feels the fixtures over the Christmas period are unfair, with Manchester United among the clubs punished.