Brett Pitman took only three minutes to make his mark on his return to the Robins starting line up when he got on the end of Jon Stead's cross for his sixth goal in as many games.
Lee Johnson extended the lead just before half-time when he steered in another Stead cross.
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.
Source: PA
Source: PA