The newly launched start-up won the account after a pitch involving M&C Saatchi, The Red Brick Road and Leagas Delaney, which was handled by the AAR.
The agency will work on a campaign centred on press and billboard advertising, with some TV activity.
The value of the account in terms of billings is around £5m, but sources said it could rise to over £7m.
Adam & Eve was launched by the former RKCR/Y&R senior management team of James Murphy, Ben Priest, David Golding and Naked's Jon Forsyth.
Westfield, which is Australian-owned, recently awarded the media account to Havas-owned agency Arena BLM.
The shopping centre will include four main stores, around 265 speciality shops, restaurants, a cinema, a health and beauty retreat and a gym.
The new centre has a catchment area of 2.7m people, with an estimated combined retail spend of £5bn.
It is understood the appointment does not affect Beattie McGuiness Bungay (BMB), which already handles creative duties for Westfield's regional shopping centres including Westfield Derby.
Westfield did not return calls at the time of writing.