But the resurgent Millers weathered an early City storm and went in front just seconds after the blow of losing midfielder Paul McLaren in the 26th minute.
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Six minutes from the break Shaw netted a superb predator's goal when he forced his way past defender Liam Fontaine - City claiming he used a hand - and crashed the ball past keeper Adriano Basso from the edge of the box.
It could have been worse for Bristol as Shaw also saw a superb half volley saved by Basso before Lee Williamson also tested the keeper with a stinging shot.
City had a handful of chances, but Steve Brooker, Dave Cotterill and Scott Murray failed to make the most of them.
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Bristol should have pulled one back five minutes later when Brooker blasted a 15-yard effort straight at keeper Gary Montgomery, while they hit the net in a late surge as Cole Skuse pounced on a loose ball in the box following Montgomery's save from Mark McCammon.

