City keeper Adriano Basso made his first save of the game inside two minutes to keep out a header from Chris McCann, while the same player shot wide after being well found by Chris Eagles.
Eagles almost opened the scoring but volleyed a right-foot shot against the crossbar.
City fired a warning shot of their own when Lee Johnson played fired the ball across the face of the six-yard box.
Striker Dele Adebola created an opportunity on 35 minutes but his cross was turned wide by Nicky Maynard.
The home side took the lead their play merited five minutes before half time when Wade Elliott was pushed over inside the penalty area by Jamie McAllister and Graham Alexander converted the spot-kick.
The Clarets increased their lead three minutes later when a Robbie Blake corner found the head of Steven Caldwell and the ball was volleyed into the net by the left foot of Elliott thanks to a slight deflection.
Basso did well to deny Elliott six minutes into the second half after diving to his left to keep out a strike from the Burnley midfielder.
City boss Gary Johnson made a double change but his side struggled to get a foothold into the game which was made safe for the home side after 73 minutes when a cross from Eagles was handled by Bradley Orr and Alexander converted his second spot-kick.
The scoring was completed six minutes from the end when Basso parried a shot from Eagles and substitute Joey Gudjonsson scored from close range.