Giroud leaves Arsenal having played in their 3-1 defeat to Swansea on Tuesday night, coming off the bench in the latter stages of the game.
The French striker has now signed an 18-month contract at Stamford Bridge, where he will provide much-needed competition for the struggling Alvaro Morata.
Giroud has found it hard to win his place in the Arsenal starting XI this season, with manager Arsene Wenger preferring to start Alexandre Lacazette in his place.
Pierre-Emerick Aubameyang's arrival to the Emirates has forced the 31-year-old, who was keen to stay in London despite Borussia Dortmund links, out of the club.
For Chelsea, it is a positive end to their search for a new striker.
Their move for West Ham's Andy Carroll was scuppered through injury, while they also considered moves for Ashley Barnes, Peter Crouch and Edin Dzeko.
Giroud was a surprise inclusion in Arsenal's squad for the game against Swansea given the transfer rumours.