AMV pitched against AKQA, Tullo Marshall Warren, Big Mouth Media and the incumbent on the business, McCann Manchester.
AMV handles digital communications in-house in an operation headed by Charlie Cannell. Its i-Chameleon digital venture was merged back into the main agency, dropping the brand name, several months ago.
The agency already handles Sainsbury's Finance above the line, while TMW handles its direct marketing account.
AMV will now be briefed with creating digital advertising campaigns, as well as web build projects, for the company's seven core products.
Jan Cutting, the head of customer acquisition at Sainsbury's Finance, said: "We are very excited by the ideas presented by AMV." Farah Ramzan Golant, AMV's chief executive, said: "This is a great win for us against big competition."