The former Sampdoria striker will ply his trade under Massimiliano Allegri as of the coming season, while his former Blucerchiati team-mate has moved in the opposite direction
Reports in Italy have also suggested that the Rossoneri have paid their city rivals an additional €7 million for the transfer to go through.
Both clubs already reached an agreement over a swap on Monday evening, with both players undergoing medicals at their respective new clubs the following day, and the deal has now been made official.
Milan were in the market for a new targetman following the departure of Zlatan Ibrahimovic to Paris Saint-Germain, and head coach Massimiliano Allegri has now got his man in Pazzini.
The 28-year-old was no longer needed at Inter as Andrea Stramaccioni did not see the forward in his first-team plans, and he will now continue his career with the red side of San Siro.
Cassano, meanwhile, filed a transfer request as he felt Milan showed a lack of ambition by letting go of several key players this summer, and has jumped at the opportunity of joining the club he supported as a child.
"Finally, after an eternity, I have ended up at the team that I support," he told Inter Channel. "A big hello to the fans, who are Interisti, just like me!
"This is very important to me because when you support a team then end up signing for them, it means you've reached the top.
"Now I'm just focused on performing well right from the start against Pescara. I'm going to wear the No.99 shirt - it's a number which means a lot to me."
The Italy forward, 30, has signed on with Inter until the summer of 2014 and was welcomed by Stramaccioni and captain Javier Zanetti on the training pitch on Wednesday morning.