The London-based practice, which was featured among last year's AJ 40 under 40 starlets, was chosen ahead of David Morley Architects, Lynch Architects and Sanei Hopkins Architects.
The prize-winning project includes plans to revamp the school's existing pavilion and create a new two-storey extension which together will provide changing space for 152 people and seating capacity for 200.
Describing the scheme, practice director Richard Scott said: 'Reflecting at length from the terrace of the existing sports pavilion, we have imagined 100 years of sporting endeavour and achievement.
'We were inspired by the honours wall in the Long Room and have realised the need for the building to have an aspect of record and renewal.
He added: 'The extension scheme flows out of the existing building strata rather than being set apart through juxtaposition.'
Bedford's selection committee was headed by Eric Parry who acted as a consultant throughout the competition.