Guinea international Bangoura, who cost the Potteries outfit £900,000, struck after 16 minutes and Sidibe added a second five minutes from the end to condemn Wednesday to their fourth outing without a win.
Sunderland striker Daryl Murphy, who arrived on loan at Hillsborough this week, almost marked his debut with an early goal when he burst through after five minutes, but Steve Simonsen parried his shot and ex-Wednesday defender Marlon Broomes cleared.
Wedneday winger Chris Brunt forced another save from Simonsen when he lashed a 25-yard free-kick that curled in dangerously.
But Stoke were ahead after 16 minutes when a long clearance from Carl Hoekens wrongfooted the Wednesday defence and Bangoura kept his head to slot the ball past keeper Nicky Weaver.
Two minutes later Bangoura could have added a second as Wednesday's sloppy defending let in the striker again, but Weaver made an impressive point-blank save.
Wednesday were reduced to shooting from distance as Stoke protected the lead, but two more long-range efforts from Brunt were wayward before Paul Heckingbottom headed a corner just over and Murphy found his way through only to see his shot saved by Simonsen.
Simonsen was in impressive form in the Stoke goal pulling off an impressive stop as Brunt threatened to level the score six minutes into the second half.
And as the home side threatened again five minutes later Simonsen saved again after Glenn Whelan produced a sharp shot.
Murphy was forced wide as he broke down the left and his angled shot failed to test the keeper.
When Stoke hit Wednesday on the break only another reaction save from Weaver denied Broomes as he stretched to stab a cross goalwards.
Sidibe struck in the 85th minute driving in a first time shot from six yards after Paul Gallagher hit a low cross from the right.