The 20-year-old has moved to San Siro on loan until the end of the Serie A season, with Milan having secured an option to then sign him permanently.
Saelemaekers has caught the eye for Anderlecht and is an under-21 international whom the Belgian club believes has a bright future.
Milan announced on its website that Saelemaekers would meet his new team-mates on Saturday before being presented at a news conference next week.
Anderlecht sporting director Michael Verschueren said: "Alexis caught the eye of a great club this season. It's wonderful for him and it's also a good sign for our training of youngsters.
"At his age, it's a dream you couldn't help but want to experience.
"I understand that fans would have preferred him not to go. But that's football today. When the big clubs like AC Milan knock on the door, you have to do the right thing."
Milan, which welcomed Zlatan Ibrahimovic back to the club in January, is on a high after winning its past three league games to climb to eighth in Serie A.
It also reached the Coppa Italia semi-finals with victory over Torino on Wednesday (AEDT)
Verschueren said: "Milan conducted themselves very well in this matter with Anderlecht, so I think there are no losers.
"We obviously wish Alexis the best and we're proud to have coached him here."