Hired by chief executive Trevor Bish-Jones three years ago, Robertson was responsible for the appointment of Bartle Bogle Hegarty, which created the Wooly and Worth ads. His previous roles include marketing director of B&Q and WH Smith.
Despite several big-budget campaigns, the retailer's sales have failed to pick up and in January it announced a 4.1% drop in like-for-like sales.
Robertson's position was placed in doubt at the end of last year when Bish-Jones brought in Asda non-food director Tony Page as board-level sales and marketing director.