Josh Brownhill’s first-half header put the Robins ahead before Famara Diedhiou doubled their advantage.
Aboubakar Kamara pulled one back late on but Lee Johnson’s side held on for the win.
In the early kick-off, Leeds moved to the top of the table for 24 hours at least after getting the better of Yorkshire rivals Huddersfield with a 2-0 win at the John Smith’s Stadium.
Goals from Ezgjan Alioski and Pablo Hernandez in the second half means Marcelo Bielsa’s men leapfrog West Brom, who host Swansea on Sunday, in the table.
Two goals from Eberechi Eze gave Queens Park Rangers their first win since October as they beat Preston 2-0.
The England Under-21 star opened the scoring in the 16th minute and then added a second from the penalty spot on 67 minutes.
Meanwhile, Charlton’s winless run stretched to eight matches after they suffered a 1-0 away loss to Middlesbrough, George Saville breaking the deadlock with less than a minute gone at the Riverside.
Jarrod Bowen netted twice in quick succession as Hull City came from behind to beat Stoke 2-1 and inflict a third successive loss on the Potters.
A stoppage-time winner from George Moncur helped Luton Town beat Wigan 2-1 as they bounced from their 7-0 thrashing to Brentford last week.
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