Report from DSG
Brett Pitman took only three minutes to make his mark on his return to the Robins starting line up before Lee Johnson extended the lead just before half-time.
Jamal Campbell-Ryce put the game beyond Cardiff early in the second half to edge Keith Millen's side away from the relegation zone.
Meanwhile, the Bluebirds have won only one of the last seven league matches and have slipped to fifth in the table.
Pitman was the only change to the Robins side that drew with Crystal Palace.
Chris Riggott and Craig Bellamy returned to the Cardiff team after being rested during the defeat at Watford, while Tom Heaton replaced the unwell David Marshall in goal.
Pitman picked up where he left off on Tuesday, when his injury-time penalty earned the Robins a point. He continued his recent goal scoring form by netting after just three minutes.
Pitman laid the ball off to strike partner Jon Stead, spun into the box and got on the end of Stead's cross to beat Heaton from close range for his sixth goal in as many games.
Cardiff responded well and David James made a save to keep out Peter Whittingham's 25-yarder before Michael Chopra and Mark Hudson shot over the crossbar.
Chopra then struck a post with an effort from the edge of the box just after the half hour mark.
In between, Heaton blocked shots from Pitman and Campbell-Ryce but there was little the former Manchester United trainee could do to prevent his side from going two goals behind after 39 minutes.
Stead again provided the assist. Campbell-Ryce dummied his low cross at the near post and Johnson steered the ball into the bottom corner from 12 yards out.
The Robins continued to look threatening at the start of the second half and Albert Adomah went close before Heaton tipped a shot over from Pitman.
It was merely delaying the inevitable and Campbell-Ryce extended the lead in the 55th minute.
The winger ran deep into the Cardiff half as the Bluebirds defence backed off, he took on Mark Hudson and found the bottom corner with a right-foot shot from the edge of the area.
Stead and Adomah went close to adding further goals just after the hour mark.