The global account had been previously handled by O&M from its Paris office. The WPP-owned agency also held the UK business.
Global vice-president of marketing for Kraft, Daryl Fielding, is understood to have led the pitch for the business.
A spokesperson for Kraft said: "We have completed extensive work to help us understand the Milka consumer: what they want in life, what they are looking for from the brand and how cultural differences affect the perspectives and the way in which they interact with the brand."
Following the takeover, Kraft is now looking to move the Milka brand toward the premium chocolate market. It currently competes with a range of Cadbury brands including Dairy Milk and Flake, the creative accounts for which are held by Fallon and Saatchi & Saatchi respectively.
Ogilvy has recently launched a series of TV ads for Milka featuring cows, which allied the brand to its Alpine roots.
Crispin Porter + Bogusky already handles the Kraft business for its Macaroni & Cheese range, which it won in March.