Cairns, who is also a group account director on BSkyB, is important to HHCL's hold on the £60 million business.
The agency made Cairns a counter-offer on Tuesday, after she had signed a contract with BMB and tendered her resignation.
Although Sky has not signalled any intention of reviewing its creative account, both BMB and Clemmow Hornby Inge are positioning themselves to challenge the business should it decide to move. In September, Nick Howarth, the HHCL chief executive and lead account manager on the Sky account, was poached by CHI, where he has taken up the position of group chief executive.
Cairns has been at HHCL since 2000. She joins BMB a month after it poached the £4.5 million Sky One account from Wieden & Kennedy in a shift of creative duties.