The home side, playing with a strong breeze at their backs, had much the better of the opening 45 minutes.
Although the flow of the game was disrupted somewhat by the wind and some fussy refereeing decisions, Stoke played throughout with a good tempo.
They took the lead after 14 minutes when Liam Lawrence was fouled just outside the penalty area and he got up to take the free-kick himself. His perfectly flighted ball was headed powerfully past Adriano Basso from close range by Sidibe.
Stoke continued to be quicker to the ball and doubled their lead after 36 minutes when a long clearance was played on by Ricardo Fuller for Sidibe to sprint clear and calmly stroke the ball past Basso as the keeper came out.
Carlo Nash in the Potters goal was not troubled and Stoke were good value for their 2-0 lead.
Clearly disappointed by his side's performance, Bristol City manager Gary Johnson made a double substitution at half-time, bringing on Lee Trundle and David Noble.
Trundle almost made contact with a cross at the far post straightaway but, after an hour, Stoke's Sidibe had a shot cleared off the line while Lawrence and Fuller also went close as Stoke turned up the pressure.
But after 67 minutes Bristol City scored a goal out of the blue with their first corner of the game when Michael McIndoe's flag kick was headed home by Dele Adebola.
The visitors then came alive and put plenty of pressure on the Stoke defence.
Stoke were forced onto the back foot, with McIndoe starting to cause problems. Ryan Shawcross had to clear the ball off the line but the defence held firm to take the vital three points and hoist the side into the second automatic promotion spot.