The veteran Frenchman, 34, will wear the No9 shirt for Milan after calling time on his spell at Stamford Bridge, which saw him score 39 goals in 119 appearances.
He lifted the Champions League, Europa League and FA Cup with the Blues but never managed to establish himself as a regular starter, regardless of who was manager.
Giroud, who joined Chelsea from Arsenal for £18million in January 2018, took to Twitter yesterday to confirm he was moving on.
He said: "I'm starting a new journey with a light and happy heart. To all the Blues, to my team-mates, to all my coaches, to the whole club, a huge thank you for these special moments."
The French international is the second Chelsea player to head to the San Siro on a permanent basis this summer after centre-back Fikayo Tomori signed following a successful loan spell.