Fallon beat the design companies Lambie Nairn, Wolff Olins, The Partners and Red Bee Media, the former BBC in-house design department, to the task.
The pitch is understood to have been led by Mark Elwood, the Fallon head of art.
The rebrand is in addition to Fallon's existing BBC Radio advertising brief, under which it creates TV print and digital advertising for the corporation's stations.
Fallon has created a number of iconic ad campaigns for BBC radio stations. It created the launch campaign for the digital urban music station BBC Radio 1Xtra.
A recent BBC radio promotion campaign was a DFGW-created TV spot for Radio 2 that brought Elvis Presley back to life, complete with a backing band including Jimi Hendrix on guitar and The Who's Keith Moon on drums.
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