The practice beat David Chipperfield Architects, Wilkinson Eyre, Bennetts Associates, Rick Mather and Stanton Williams Architects to win the high-profile commission.
A completion date for the £80 million project has been set for 2011, and it is thought the new exhibition space will transform many of the museum's facilities and enable it to develop and stage large-scale touring exhibitions.
It is yet another feather in the cap for Rogers, who is showing no signs of slowing down.
Earlier in the year his practice was chosen to lead the £908 million Chelsea Barracks scheme, and it is already working on One Hyde Park - set to become the most expensive residential scheme in the country.
Rogers was also handed the Pritzker Prize in March, which capped an impressive 12 months for the architect - also picked up the Stirling Prize for his Barajas Airport last November.