The Potters were locked on course for a solitary point and mid-table mediocrity before super-sub Sidibe netted with only ten minutes left and Liam Lawrence added the gloss in injury time to keep the play-off dream alive.
A disruptive gale billowing through the Britannia Stadium reduced the opening 15 minutes of play to an exhibition of pub football, but as conditions improved a fine first 45 minutes of football unfolded.
Leicester were the first beneficiaries of the settled conditions when Danny Tiatto arrowed over a right-flank free-kick which was somehow beyond Steve Simonsen's reach.
The City keeper appeared to be shoved as he came to collect, but all arguments were drowned out by the celebrations as Darren Kenton slotted into the vacant net.
But the lead was short-lived when a neat approach from the hosts opened up the Foxes.
City flyer Ricardo Fuller drew the escort of two Leicester defenders before dropping a pass into the path of Lawrence, who picked out the advancing Jon Parkin to slam in his first goal for the Potters.
Fuller was proving too hot to handle for the Leicester rearguard and he infiltrated the enemy line again on the 28th minute drawing a wild lunge from Kenton.
The speedy Jamaican dusted himself off before driving his side into a deserved lead from the resultant penalty.
Despite Fuller's best efforts to be the star of the show, Stoke were left indebted to the cat-like heroics of Simonsen for their half-time goal advantage.
The super-stopper first beat out Tiatto's point-blank rocket before flinging himself across his line to turn Jason Jarrett's goal-bound effort around his right-hand post.
But the gutsy Foxes only needed 13 minutes of the second half to claw a way back.
A lightening break had Stoke out-numbered and left Shaun Newton to drop a ball into Elvis Hammond to stroke in his fifth of the campaign.
The Leicester contingent would have been forgiven for thinking they had done enough to earn a share of the spoils until Sidibe applied the jump-leads.
The Mali international coolly slotted in Fuller's return ball from point-blank range to draw a line under a barren spell of 11 games without finding the target.
And all was forgiven for his recent goal slump when some gutsy play by the corner flag earned his strike partner an eye on goal when some would have simply ran down the clock.
The towering target-man turned his marker before finding Fuller whose sliced shot fortuitously landed in the lap of Lawrence at the back-post and he made no mistake.