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Chesterfield were given the chance to level the scores in the 25th minute after Derrick Williams gave away a penalty for bringing down Doyle inside the box.
Doyle himself stepped up to take the spot kick and made no mistake by firing the ball in straight down the middle.
Bristol almost regained their lead two minutes later, but Bryan's curling effort hit the outside of the post.
A minute after the break Bristol scored their second of the match through Williams, who headed in following a corner.
However, the visitors drew level for the second time just before the hour mark when Doyle slotted in from close range following Tendayi Darikwa's cross.
Chesterfield were then inches away from taking the lead as Luke Freeman's drive crashed against the bar.
In the third minute of stoppage time Bristol claimed the points as Burns met Elliott's cross before smashing the ball past Lee on the volley.