The Shakers made the perfect start when midfielder Danny Mayor was allowed to run at the Robins defence unchecked and he fired home from 20 yards.
The visitors continued to have the better of things with Danny Rose hitting the side-netting and Nicky Adams heading wide from in front of goal.
And Bury’s dominance was rewarded when they doubled their lead after 17 minutes. Andrew Tutte got behind the Cheltenham defence on the right and when he drilled in a cross, Rose was on hand to chest the ball over the line from two yards out.
Cheltenham finally came to life when, after 24 minutes, Wes Burns won the ball midway inside the Bury half and played in Denny Johnstone who finished well from 18 yards.
Burns had a chance to level just before the break but he sent his shot over the crossbar. The on loan Bristol City forward had another opportunity early in a lacklustre second half but he was just off target.