The Owls were ripped apart defensively and trailed three nil at the interval following an own goal from Joe Wildsmith, Bobby Reid and a Lee Tomlin penalty.
After the restart, Jonathan Kodjia also registered from 12 yards before substitute Ross Wallace bagged a consolation.
Victory saw Wednesday fall to sixth, the final play-off position, as a result of Derby County's comprehensive win over League One bound Bolton Wanderers.
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It was a desperately disappointing showing by the Owls and you could only feel sorry for the 1,400 plus travelling fans who had made the long journey south.
Boss Carlos Carvalhal shuffled his pack, making five changes to the starting line-up. In came Liam Palmer, Michael Turner, Aiden McGeady, Barry Bannan and Atdhe Nuhiu. Jack Hunt, Glenn Loovens, Wallace, Lucas Joao and Gary Hooper dropped out.
Loovens (ankle) and Hooper (foot) missed out due to injury.
Wednesday were forced to make a late change to their substitutes bench. Originally in their match-day squad, they named six loan players: McGeady, Turner, Daniel Pudil, Vincent Sasso, Alex Lopez and Joe Bennett.
However, Football League rules state a "maximum of 5 loan players (Standard, International, Emergency or Youth) can be named in the players listed on a team sheet for any individual match played under the auspices of The League". It meant Bennett dropped out and Hunt took his place on the bench.
If that wasn't bad enough, Keiren Westwood picked up an injury during the warm up, leading to Joe Wildsmith being drafted in between the sticks.
Wednesday made all the early running and could have opening the scoring through McGeady after the Republic of Ireland international caught Bristol City's backline napping. The winger broke clear only to be foiled superbly by boyood Wednesday fan Richard O'Donnell.
Bannan, recalled to the starting line-up after serving his three-match suspension, curled a 25-yard free kick narrowly wide as Carvalhal's men continued to dominate proceedings.
The hosts made Wednesday pay for their profligacy, breaking the deadlock courtesy of Tomlin, who volleyed home through a crowded penalty area following a peach of a delivery by Adam Matthews.
Things got even worse for the Owls in the 13th minute as the Robins doubled their lead, Bobby Reid drilling a fierce shot past Wildsmith after a lovely cut back by Luke Freeman.
Oozing confidence, Lee Johnson's side went close to grabbing a third midway through the first period. Top-scorer Jonathan Kodjia shrugged off the attentions of Turner before unleashing a rasping drive which Wildsmith fumbled but, fortunately for the visitors, captain Korey Smith blazed over the follow up.
Wednesday looked rattle and too many individuals had not turned up. There was a sloppiness and lack of sharpness to their play with and without the ball.
Johnson's charges continued to threaten in the final third and opened up Wednesday's rearguard again after a wonderful counter attack involving Kodjia, Reid and Freeman. It took a smart stop by Wildsmith to thwart Freeman.
It was one way traffic and Wildsmith produced a good stop to prevent Marlon Pack from getting on the score-sheet
Moments after Reid had a strong penalty appeal waved away after a collision with Tom Lees, the Robins were awarded a spot-kick after Palmer upended Tomlin. The forward picked himself up and coolly converted from 12 yards, despite Wildsmith guessing the right way.
Carvalhal, unsurprisingly, made a double alteration at the interval, throwing on Lucas Joao and Ross Wallace.
But the changes did not have the desired affect and City claimed a fourth in the 53rd minute from the penalty spot after Wildsmith brought down the lively Reid. Kodjia stepped up to compound Wednesday's problems.
Against the run of the play, the Owls reduced their arrears on the hour mark. Wallace's well-flighted free kick hit the post and went in off O'Donnell's back.
That briefly slowed down the Robins momentum but Kodjia should have tested the reflexes of Wildsmith after gliding past the hapless Turner only to drag his effort wide.
Wednesday kept on battling and Wallace bent a free kick off target before Forestieri scuffed his shot from distance.
Even in the closing stages, the visitors carved out opportunities as Forestieri toe-poked wide before Marco Matias' long range strike was beaten away by O'Donnell.