Though the formal press conference has been scheduled for Tuesday afternoon, Milan posted a picture of the Ghana international shaking hands with Milan vice-president Adriano Galliani on Twitter after the deal was done.
The midfielder's move away from Stamford Bridge had been shrouded with doubt over the weekend by suggestions he had failed a medical after arriving in Italy on Friday afternoon.
According to reports in the Italian media, the 31-year-old failed tests related to an old knee injury.
But Essien later said that his medical would not scupper the switch.
"Haters will broadcast your failure but whisper your success.Medicals done, looking forward to start my new life with AC Milan," Essien posted on his Twitter page.
He has two Premier League winners' medals as well as four FA Cups and the 2012 Champions League to his name during his time at Chelsea.
His time in London came either side of a brief loan spell at Real Madrid last season, where current Blues boss Jose Mourinho was working at the time, and before Mourinho recruited him to Chelsea in 2005 he appeared for Bastia and Lyon.
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.
Manager Jose Mourinho told his club's website: "To show Michael respect is to not stop him going.
Ahead of his presentation tomorrow, the 31-year-old spoke to Milan’s in house television channel on the experience so far.
“Thank you for the welcome, everything is making for a beautiful impression,” responded Essien on Milan Channel.
“Milan is a great club and I am happy to be here. I’m very happy for the words from Seedorf, he is a good person and I am happy to be here.
“And it is a pleasure to have heard his words [of support]. I await tomorrow, then there will be a presentation and afterwards I begin to train.
“I cannot wait to meet the team and the Milan family - from there we will see what happens.
“I always give the most, I fight for the team from the first minute. I hope to achieve something.
“Objectives? We shall see, now I am just waiting to start playing and to do my best for Milan.”
"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."