The striker has been in the Premier League since 2012
The World Cup-winning France striker teased the Milan switch by revealing he would be embarking on “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,” Giroud posted on twitter.
“I’m starting a new journey with a light and happy heart.
“Our victories in the FA Cup, Europa League and Champions League have been magnificent.”
The ever-dependable Giroud bagged a goal in Chelsea’s 4-1 Europa League final win over his former club Arsenal in the 2019 final in Baku.
And the France hitman then played a crucial role in Chelsea’s run to the 2021 Champions League crown too.
Giroud became the oldest player to net a Champions League hat-trick when claiming all the goals in the Blues’ 4-0 win at Sevilla in December 2020.
Giroud’s west London exit is expected to be followed in the coming days by defender Marc Guehi’s move to Crystal Palace.
The Eagles are understood to have struck a deal to reach a fee of £18million to convince Chelsea to part with the highly-rated 21-year-old, who has spent the last season and a half on loan at Swansea.