Pazzini moves in the opposite direction as part of the deal, which involves an undisclosed fee going to the Rossoneri.
A statement on Inter's official website read: "The nerazzurri have signed the Bari-born striker on a permanent deal from AC Milan.
"He has signed a two-year contract which runs until June 2014."
The Italy international missed a large part of last season after undergoing emergency heart surgery having suffered a stroke in October.
Inter issued another official statement which read: "Inter Milan can confirm the permanent transfer of Giampaolo Pazzini to AC Milan.
"Pazzini, who scored 19 goals in 60 appearances for the Nerazzurri, leaves with the best wishes of everyone at the club."
Inter president Massimo Moratti told the Gazzetta dello Sport on Tuesday: "Signing Cassano is an interesting move. I hope he can do well.
"He's an interesting player and I hope he'll be useful. We'll be very happy if Cassano brings his enthusiasm with him."
The 30-year-old began his career with hometown club Bari before making a name for himself at Roma.
He then became only the second-ever Italian player to sign for Real Madrid before returning to Italy with Sampdoria, originally on loan, in 2007.
His successful spell at the Stadio Luigi Ferraris preceded a move to the San Siro in 2011.
Cassano later told Inter Channel: "After what has seemed like a lifetime I've joined the club that I support.
"That's important to me, if you're a fan of a club you give your all.
"It will put a smile back on my face.
"Will I score on my debut? The important thing is being ready in time for the Pescara game.
"I'll wear shirt number 99, as promised by (directors Marco) Branca and (Piero) Ausilio."