The 31-year-old joined the Blues from Lyon in 2005 and made 256 appearances for the club, also enjoying a season on loan with Real Madrid in 2012-13.
His first-team opportunities at Stamford Bridge had been limited in the current campaign and he has therefore now opted to pursue a career in Serie A.
Milan published a photograph of Essien posing with Milan vice-president Adriano Galliani on their official Twitter account, while Chelsea announced the deal on their official website.
Manager Jose Mourinho told his club's website: "To show Michael respect is to not stop him going.
"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."