The 31-year-old Ghana international underwent a medical at Milan before putting pen to paper on a contract that ties him to the Italian giants until June 30, 2015
Essien will now link up with his new team-mates on Tuesday and he admits he is itching to begin life with the seven-time European champions.
He tweeted: "I have just signed my contract with one of the best clubs in the world. Im very happy and looking forward to winning with the team#ForzaMilan."
"I cannot wait to join the team in training and be a part of Milan's family," he added on Milan Channel.
Essien joins a Milan side who have fallen on hard times of late with their only major trophy success in the last six years being the 2010/11 Serie A title.
This season they are struggling down in ninth place in the Serie A standings, 28 points behind league leaders Juventus, while they were also knocked out of Coppa Italia quarter-finals at home by Udinese last week.
Nevertheless, Essien is excited to be joining such a "great club" and has pledged to give new Rossoneri coach Clarence Seedorf his all.
"I am very happy to have spoken to Seedorf, he is a nice person," he said. "It's a pleasure to have heard what Seedorf had to say about me.
"I always give my best and battle for the team from the first minute until the last.
"I hope to achieve something important and give my best for Milan."
Essien, who could make his Serie A debut against Torino at the San Siro on Sunday, joined Chelsea from Lyon in 2005 for £24.4million and made over 250 appearances for the Blues.
However, he had found first-team opportunities at Stamford Bridge limited after returning from his loan spell at Real Madrid last season and decided to call time on his stay with the Blues.
Essien goes with Chelsea coach Jose Mourinho's blessing, although the Portuguese admits he is sorry to see the versatile player leave.
"To show Michael respect is to not stop him going," Mourinho said on his club's official website.
"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 is Milan's third reinforcement of the winter transfer market following the arrival of Japanese playmaker Keisuke Honda from CSKA Moscow and French centre-half Adil Rami from Valencia.
Despite their struggles in the first half of the season, which led to the sacking of Massimiliano Allegri earlier this month, Milan still have plenty to play for this campaign.
They are only five points off the European places in Serie A and are still alive in the Champions League, where they take on Atletico Madrid in the last 16 of the competition next month.
Source : PA