It will develop a range of campaigns to improve public understanding and confidence in the Red Cross. These include emergency response in the UK and overseas as well as first-aid training and other services in local communities in this country.
Direct MediaCom handles the Red Cross' media and is unaffected by the review.
The pitch was led by Phil Talbot, the Red Cross director of communications.
He said: "DFGW showed great understanding and empathy with our organisation and presented us with powerful ideas that have potential for all channels."
Tom Vick, the joint managing director of DFGW, added: "It's a great honour to be working with one of the most- recognised brands in the world. We're really looking forward to helping Phil and his team galvanise and inspire the British Red Cross."