Cassano, 30, was at training with Inter and Pazzini, 28, with Milan on Wednesday morning and, following successful medicals, the Rossoneri announced the news on their official website.
A statement read: "AC Milan announces that it has acquired outright from FC Internazionale Giampaolo Pazzini and simultaneously has sold Antonio Cassano outright to FC Internazionale."
It is believed that Milan have paid Inter a fee in the region of €7 million as part of the deal, which came to pass after Cassano had grown disenchanted with the club after seeing the likes of Zlatan Ibrahimovic and Thiago Silva depart.
Inter welcomed their newest signing with a brief message on their own site. It read: "Today is the first day of Antonio Cassano's adventure at Inter. This morning the striker arrived at the Centro Sportivo Angelo Moratti, where Andrea Stramaccioni and captain Javier Zanetti were waiting to welcome him."
Pazzini's new teammate, and long-term friend, Riccardo Montolivo was full of praise on Tuesday for his side's capture of the striker.
"As a person he is splendid and we have been friends for many years," he told Milan Channel. "Together we won the Coppa Italia Primavera with the Atalanta youth team, playing together for that club, at Fiorentina, in the Under-21 Italy side and for la Nazionale.
"It's wonderful to meet up here, at the club with the most trophies in the world, especially as I feel our squad is missing a player exactly like him."