The Gunners agreed to sell Giroud to their London rivals after wrapping up a club-record £56m deal for Pierre-Emerick Aubameyang. Dortmund were prepared to sell Aubameyang once they knew they had a deal secured for Michy Batshuayi.
Giroud completed his medical on Friday and agreed personal terms for his move across London.
The signing of Giroud ends Chelsea’s frantic search for a striker this January, with the club chasing Roma’s Edin Dzeko, Andy Carroll from West Ham, and even 37-year-old Stoke star Peter Crouch.
Wenger hailed the ‘top class’ commitment of Giroud after the striker played in Arsenal’s defeat by Swansea despite being on the brink of joining Chelsea.