Coventry took an early lead through Franck Moussa, who charged into the box before seeing his shot go under the body of Cardiff City loanee Simon Moore in goal.
Adam Barton came close to doubling the visitors' advantage, though his 25-yard effort was tipped over the bar by Moore to keep the scoreline at 1-0.
Joe Murphy had to ensure Steven Pressley's side retained their two-goal lead after being called into action by Gregg Cunningham. The defender picked up a through-ball before seeing his attempt saved by Coventry's keeper.
Murphy was almost caught out shortly after when he was spotted off his line by Sam Baldock, though the forward's finish from near the corner flag came back off the post.
The home side found a route back into the game with less than 10 minutes remaining thanks to Wes Burns header, which took a deflection on its way past Murphy to give Bristol a lifeline.
Despite pushing for an equaliser, the Robins failed to break Coventry down, and remain just one point above the relegation zone.