Birsa, 27, is a versatile attacker and joins the club on a three-year deal following the sale of Kevin-Prince Boateng to Schalke.
The Slovenia international has been capped 62 times for his country and spent a season on loan at Torino last year, having failed to establish himself as a regular first-team member at Genoa.
Meanwhile, full-back Antonini - who was linked with Paolo Di Canio's Sunderland - has moved in the opposite direction, also signing a three-year deal.
The 31-year old was loaned out to a series of clubs during his time with the Serie A outfit, including Sampdoria, Modena and Siena.
He became a particular favourite with coach Massimiliano Allegri, who installed him as first choice left-back for the 2010-11 season, where he helped his side to their 18th league title.
In an emotional tweet, he told fans: "After 23 years in Rossonero, today I salute the team I support. The time has come for me to go. Goodbye my Milan, I gave my heart for you. Thank you for everything."