The Ghanian found the top corner from 18 yards before Steven Davies added a second shortly after the break as City continued their unbeaten run at Ashton Gate since Sean O'Driscoll took charge.
Boro, without a win on their travels since early December, struggled to create chances throughout as they slipped further behind the play-off places.
The hosts took the lead just after the half hour mark though with one of the best goals the strugglers have scored this season.
Neil Kilkenny's raking 40-yard pass found Adomah and after taking on George Friend he cut inside and curled the ball into the top corner via the inside of the post for his fifth goal of the season.
It only took seven second-half minutes for the Robins to double their lead when Davies was slipped in by Stephen Pearson after a quickly taken free-kick and he lashed the ball home from an angle to delight the Ashton Gate faithful.