Sidibe: At the double for Potters
Dele Adebola pulled one back after the break but there was to be no equaliser for the Robins, who now look set to fight it out in the play-offs.
Sidibe was in the side at the expense of Shola Ameobi but it proved to be a tactical master stroke by Tony Pulis as the visitors' defence struggled to deal with his pace and power.
Both sides knew before kick-off that a win would send them second behind West Brom and into the automatic promotion places, and Stoke are now assured of a play-off spot at the very least.
Adebola had an early effort for the Robins from 20 yards but Carlo Nash in the Stoke goal produced a comfortable save.
Stoke went ahead in the 14th minute when Sidibe headed in Liam Lawrence's beautifully flighted free-kick from the right, which had been awarded when Jamie McAllister was penalised for an innocuous-looking challenge on the former Sunderland midfielder.
Four minutes later Ricardo Fuller could have doubled the home side's advantage but he skied his shot from the edge of the box.
The home side were enjoying the lion's share of possession and Sidibe doubled Stoke's advantage 10 minutes before the break when he latched onto Fuller's flick-on and burst clear before coolly slotting the ball home.
Brian Wilson, a former trainee at the Britannia Stadium, smashed a 30-yard free-kick high and wide in the closing moments of the first half as the visitors struggled to create a meaningful opening against a well-organised Stoke defence.
Michael McIndoe's 49th-minute cross caused panic in the Stoke defence and only a fine stop by Nash prevented Lee Trundle pulling a goal back.
McIndoe tested Nash with a half-volley from the edge of the box, but again the former Preston custodian made a comfortable save.
Sidibe was denied a hat-trick shortly before the hour mark when his shot from Lawrence's cutback was cleared off the line by Jamie McCombe.
Moments later Lawrence was a whisker away from settling the game as a contest when his 25-yard free-kick brushed the outside of the post.
The home side were rampant and the influential Lawrence found Fuller, whose shot was deflected wide off McCombe.
Lawrence then almost fluked the goal his performance deserved when his cross from the right had to be tipped over the bar by Adriano Basso in the Bristol goal.
Adebola pulled a goal back when he flung himself at McIndoe's 67th-minute corner and sent a rocket of a header into the back of the net.
After a frantic scramble in the Stoke six-yard box McIndoe picked out Marvin Elliott whose shot was brilliantly cleared off the line by Ryan Shawcross and scrambled clear.
Basso kept the away side in the game when he tipped Leon Cort's header over the bar.
Suddenly it was the visitors creating all the pressure and a series of high balls created problems amongst Stoke's backline.
But the equaliser did not materialise and in the closing stages Stoke were a constant threat on the counter-attack, substitute Ameobi skewing the best opening well wide.