The appointment follows a three-way pitch in which CubanEight was pitted against two agencies from London.
Oxfam Unwrapped manager Rick Lay said: 'We were looking for an agency that could deliver an imaginative and sustained PR campaign in the run-up to Christmas.'
The account will be headed by CubanEight co-founders Rachel Brewin and Sian Gaskell.
Oxfam Unwrapped gifts let people choose a gift, such as a goat, and send a gift card to a friend or relative as a memento. Oxfam ensures that the gift is sent to someone who needs it.
The new campaign will focus on the launch of the updated 2010 gift range and sustaining the brand profile during the run-up to Christmas.