The Ghanaian, who spent last season on loan at Real Madrid, has struggled to hold down a place in Jose Mourinho's side in England, making just nine appearances in all competitions this season.
Nemanja Matic's arrival at Stamford Bridge pushed him further down the pecking order in midfield, making a move to Italy look increasingly likely.
Essien had been linked with a move to Milan during the transfer window and confirmed on Sunday that he had passed his medical, contrary to other reports.
Chelsea confirmed the move on their website, with Mourinho wishing the midfielder well in his new challenge.
"To show Michael respect is to not stop him going," he said. "We would have been happy had he stayed because he is fantastic player for the team, but he's given everything for the club and a great chance arrived for his future.
"I know he can continue to play at a high level and I wish him success and happiness in Italy and at the World Cup in Brazil."
Essien will be unveiled as a Milan player on Tuesday, with the club having announced a press conference on their website on Monday.
The 31-year-old arrived at Chelsea in 2005 from Lyon and made 256 appearances during his time at the club, scoring 25 goals.
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