The 21-year-old midfielder, whose first year at Stamford Bridge was disrupted by serious injury, was confirmed as a target for the Italian giants on Wednesday.
Van Ginkel is understood to have joined in a season-long loan deal, though the clubs are yet to announce the specifics.
The Netherlands international underwent a medical on Monday and said that fellow countryman Nigel De Jong helped persuade him to move to San Siro.
"I am very happy. Milan is a great club where many Dutch players have played and some big names," he told the Milan Channel.
"I know Nigel De Jong from my experiences with the national team.I spoke to him about Milan and he said excellent things about the club. I will see him soon.
"I am a midfielder that defends but I also know how to score goals. I play all across the park. I want to become an important member of the squad, I am a team player and I want to meet the coach.
"I want to get game time and I think that I will have the chance to play at Milan in a great league against strong sides. This is the objective for the season.
"I haven’t as yet spoken to coach Inzaghi but I have spoken to Fernando Torres this week. He’s a good lad."
The youngster follows Torres in swapping Jose Mourinho’s side for Milan, who began their Serie A campaign by beating Lazio 3-1 on Sunday.
The Spain international striker has agreed a two-year loan with Filippo Inzaghi’s side after consistently disappointing at Stamford Bridge since his £50 million move from Liverpool.