The goal came after Chicago failed to clear the ball following a corner kick.
The Revs' Kelyn Rowe served it back into the box and Bengston was there to put a header past Chicago goalkeeper Sean Johnson from close range.
Matt Reis provided the clean sheet for New England (1-0-0), which managed just two road wins all last season.
The Fire (0-2-0) were coming off a 4-0 season-opening loss to Los Angeles.
Celebrity Families of 2014 [PHOTOS]