The agency will be responsible for the digital strategy across the entire Sure portfolio, which is known as Degree in the US and Canada, and Rexona in other markets.
R/GA will take up a place on the Unilever global digital roster, and will also work with the Impulse brand.
Previously, Unilever produced digital activity for Sure on a country-specific basis.
Mariano Sampietro, Rexona global director, said the brand saw digital "as critical to maintaining the ongoing success of the Rexona brand".
R/GA will deliver digital campaigns and long-term digital platforms for the brand in a move to strengthen the connections between the brands and their consumers.
The agency will work on the brand out of its New York, San Francisco, London, Singapore and São Paulo offices.
Iris London's relationship with Unilever is not affected by R/GA's appointment, according to an Iris spokeswoman.
The integrated agency works on Sure as well as other Unilever brands including Magnum, PG Tips, Marmite and the Unilever brand itself.
Follow Matthew Chapman at @mattchapmanUK