Rocked by an early injury to Marcus Stewart, City boss Gary Johnson threw on David Cotterill who played an influential part in the victory, while Louis Carey and Liam Fontaine had solid games.
Scunthorpe lacked discipline at the back and although manager Brian Laws switched his defence around at the interval, it failed to provide the necessary support.
The Robins were first to show when a free-kick ricocheted off the wall and David Noble drove wide, but it was Scunthorpe at this stage who looked much the livelier, with Matthew Sparrow very effective.
However, as the visitors settled down Stephen Brooker began to prove a real thorn in the side of the Scunthorpe defence, and they were often to be found in disarray and lacking in creativity.
Brooker should have opened the scoring in the 14th minute when he scuffed an easy chance and then Michael Rose almost cost Scunthorpe an own goal with a header.
Sparrow spared Scunthorpe's blushes in the 21st minute, heading off the line from Fontaine before Tommy Evans saved well from Brooker.
Cotterill lifted the ball just over the top but he made amends in the 32nd minute, clipping the ball home as Evans and Steve Foster were at sixes and sevens.
The goal had certainly been on the cards and was just reward for the Robins' endeavour.
Scunthorpe switched things round at the break and immediately went on the attack, and Billy Sharp's header was cleared off the line by Carey, before Tommy Johnson was literally a coat of paint short of the target with a ball from the left.
Johnson's skills were much in evidence when he turned brilliantly on the edge of the box to crack in a shot which visiting keeper Adriano Basso saved well.
Bristol increased their lead in the 79th minute when a static Scunthorpe defence allowed Scott Murray to race through and place the ball beyond Evans, and from there on there was never any doubt about the destination of the points.