David Ngog and Martin Petrov had opportunities to put Wanderers in front before Sidibe opened the scoring in the 25th minute of the Boxing Day clash at the Reebok Stadium in the Championship.
An excellent stop from Owls goalkeeper Chris Kirkland denied Benik Afobe 20 minutes from time and the visitors held out.
Peterborough's victory at Wolves means Wednesday remain two points above the relegation zone, five points shy of Wanderers. Dougie Freedman's pre-season promotion favourites remain slumped in 16th position following back-to-back losses.
Bolton defender Sam Ricketts made a hash of clearing a long ball forward to present Michail Antonio with an early opportunity but the Wednesday winger could not get his shot away.
The hosts proceeded to have the better of the early exchanges and Ngog - leading the line in the absence of suspended captain Kevin Davies - displayed superb close control in a fine solo run that ended with save of equal quality from Kirkland in the 15th minute.
Petrov and Ngog produced some slick interplay before the former Manchester City winger lost his footing at the vital moment 10 yards from goal.
It was a slip his team would come to regret as Wednesday took the lead against the run of play moments later.
Lewis Buxton delivered a searching cross from the right and Sidibe got the better of Tim Ream in the air to emphatically power home.
Sidibe almost had a second in the 36th minute, just failing to get the decisive touch on Rhys McCabe's corner.
Bolton were evidently rattled and might have been two behind at the break after Lee weaved his way around Ricketts and fired a shot at goalkeeper Adam Bogdan.
Ngog knew little about it when he nearly brought Bolton level in the 53rd minute - Afobe's well-struck cross from the byline following a brilliant run striking him on the chest and trickling wide.
Kirkland bettered his first-half stop from Ngog in the 70th minute when Afobe met a superb Chris Eagles cross but was denied from point-blank range.
A concerted late rally never truly materialised from Bolton and Wednesday were safe when Ngog's late header from Zat Knight's cross landed well wide.