Chelsea manager Jose Mourinho has apologised for comments he made about Everton striker Andrew Johnson after Sunday's 3-2 victory at Goodison Park.
Mourinho has changed his mind after viewing the incident on video
Mourinho claimed Johnson had dived to win a penalty and Everton responded by saying they would make an official complaint to the Football Association.
"Everton, his manager and he deserve my apologies," said Mourinho.
Everton chief executive Keith Wyness added: "Common sense has prevailed and this is an end to the matter."
Johnson said in a statement: "I am pleased that Mr Mourinho has apologised for the comments he made.
"I have the utmost respect for both Chelsea Football Club and Mr Mourinho as a manager.
606 DEBATE: Your reaction to Mourinho's apology
"Everton football club and myself can now concentrate on the games ahead. I would like to thank Everton football club, my manager David Moyes, my team-mates and my agent for all their support.
"This is now the end of the matter."
Everton initially said they were taking legal advice because the remarks questioned Johnson's "professionalism and integrity".
The incident occurred when Johnson went down after challenging Chelsea keeper Hilario for the ball.
Mourinho said that he had now watched the incident again on video and had a different view of it.
The relationship between ourselves and Chelsea has always been admirable and that, I feel certain, will continue
Everton chief executive Keith Wyness
"First I would like to say I have the utmost respect for Everton Football Club, David Moyes and their players," he stated.
"That's why I love to play them and especially at Goodison Park, where the atmosphere is magnificent.
"Secondly, after the match I was clear and said Andy Johnson is a great player and I never used aggressive words, like some managers did against my players in previous seasons, or like some others recently said about him and Ronaldo.
"I never used the word 'cheat'.
"After seeing it again on the video, [referee] Mr Halsey did wonderful work and both decisions for penalties were correct.
"Did Andy Johnson try to avoid a collision with my goalkeeper? It seems now the answer to that is yes, so Everton, his manager and he deserve my apologies."
As well as calling for an end to the matter, Everton chief executive Wyness said: "The relationship between ourselves and Chelsea has always been admirable and that, I feel certain, will continue."
A spokesman for the FA said: "We are pleased this issue has been sensibly resolved.
"It is important to remind everybody in football of the need to respect opponents and other people in the game for the good image of the sport."