The move leaves Saatchis without an executive creative director, and follows a three-year stint by Droga, who has been widely credited with raising the agency's creative profile in that time.
In 2002, Saatchis won several gold awards at Cannes, among a number of other awards, for clients including Club 18-30, the NSPCC and the MS Society.
Droga's successor has yet to be named, and Saatchis faces hiring a creative leader at a time when three other London agencies - Grey, Ogilvy & Mather and J.Walter Thompson - are also looking.
Sources say the chief executive, James Hall, and the managing director, Kevin Dundas, are not likely to make an internal appointment, preferring instead to search outside the London office, but possibly within the global network, for a replacement.
Droga started his career at the Sydney agency Omon.
After stints working in Asia, including the Saatchis Singapore office, he joined the London office in 1999 as the creative director.
It is as yet unclear when Droga will take up his new position at Publicis, or whether he will relocate to New York for the job. His hiring follows the departure of D'Arcy's worldwide chief creative officer, Lee Garfinkel, who turned down the Publicis job, instead taking a job within Omnicom's DDB Worldwide network.
Droga said: "Saatchi & Saatchi London has been the most rewarding three years of my career to date, and gives me great confidence for the new job."
The Publicis Groupe chief executive, Maurice Levy, added: "I want Publicis Worldwide to be the best creative network in the world, and I have chosen the man whom I believe is the best creative director today. Dave has a track record of amazingly relevant campaigns which win awards and market share."