Cassano, 30, has joined Inter just 18 months after he moved to the Rossoneri from Sampdoria, where he enjoyed a prodigious strike partnership with Pazzini.
Pazzini, 28, also left the Genoa club in January 2011 and the cash-strapped club subsequently went into freefall and was relegated that season.
An AC Milan statement read: "AC Milan announces that it has signed from Internazionale Giampaolo Pazzini and has sold Antonio Cassano outright to Internazionale."
Inter added in their own statement: "Today is the first day of Antonio Cassano's adventure at Inter. This morning the striker arrived at the Centro Sportivo Angelo Moratti, where [head coach] Andrea Stramaccioni and captain Javier Zanetti were waiting to welcome him."
It is believed that AC Milan have paid a fee of around £5 million as part of the deal.
Cassano's departure from Milan comes as a surprise, given that he has revived his career in his 18 months at the club and become firmly established in the Italy line-up.
"Finally I am at the team I have always supported," the former AS Roma and Real Madrid player said in a statement.
Having been overlooked for the 2010 World Cup, the 30-year-old formed Italy's strike partnership with Mario Balotelli at Euro 2012, despite missing six months of the season after undergoing minor heart surgery.
Cassano has had a colourful career marked by tantrums and personality clashes, but his early immaturities looked to have been ironed out when he joined Sampdoria from Real in 2007.
However, a verbal spat with the Sampdoria president Riccardo Garrone led to him being suspended in 2010 and he joined Milan where he became a key factor in their Serie A triumph last year.