The assignment complements Unilever's pan-European Lynx and Bertolli accounts already handled by BBH and represents a significant reinforcement of the agency's relationship with the multinational.
Unilever said the move was part of its plan to make BBH a "key strategic partner" alongside Lowe.
Matthew Bull, the Lowe chief executive, acknowledged that the termination of the agency's eight-year relationship was disappointing. Lowe successfully repitched for the business against Mother only eight months ago.
For BBH, the arrival of Surf, one of Unilever's "power brands", not only takes the agency into an advertising sector in which it has never operated before, but is the first time Unilever has entrusted it with a household name.
"Surf is a big grown-up brand and this assignment marks a coming-of-age in our relationship with Unilever," Simon Sherwood, BBH's chief operating officer, said. "This will be a big test for us and it's very important that we get it right."
Surf, launched in 1952, is the third-biggest-selling product in the washing detergent market behind Unilever's Persil and Procter & Gamble's Ariel and strives to outpace the relentless threat of Bold and Daz as well as own-labels.
"Unilever is looking for something different for Surf and clearly feels we can provide it," Sherwood added.
Lowe has been briefed to shoot a follow-up film to the ad currently airing, which features the comedians Adam and Joe putting the designer Laurence Llewellyn-Bowen through painful experiences to prove Surf's effectiveness.
The agency continues to work on Surf in other regions and handles three quarters of Unilever's global laundry business.
Bull said: "We're obviously disappointed but we're as committed as ever, if not more committed, to produce brilliant work for Unilever."