Twelve months on from losing at home to Cambridge in the same round of the competition, the Magpies came unstuck in South Yorkshire as Windass scored twice in 14 minutes to produce one of the shocks of this year's cup.
Only the woodwork denied Windass what would have been a breathtaking hat-trick as a 30-yard free-kick thundered into the crossbar, with Wednesday registering one of their best results in recent years.
Bruno Guimaraes pulled one back for Newcastle and Chris Wood missed a sitter to draw level, but it was a first defeat in 16 games in all competitions, going back to August.
Eddie Howe wanted to progress as he brought on his big guns to try and save the game, but with a Carabao Cup quarter-final to come on Tuesday and a Premier League top four race to contend with, their demise here might not be such a bad thing in the bigger picture.
For Wednesday this result, as much due to the heroics of goalkeeper Cameron Dawson as Windass' goals, continues their feelgood factor as they look to supplement their League One promotion push with a cup run.
Newcastle's first taste of what might be in store for them came when Dawson produced a fine stop to deny Alexander Isak a goal on his first appearance since September.
The Magpies created an overlap down the right and Javier Manquillo's cross was begging to be put away by the Sweden striker, but he put his header too close to Dawson, who clawed it away.
Dawson again had to be alert to deny Isak, saving a low drilled effort with his legs after another clever pass from Manquillo.
Newcastle were in control and continued to create chances as Manquillo pulled back for Matt Ritchie, whose miscued effort looped over and then Jacob Murphy could not get his lobbed effort on target after being sent clear by Ritchie.
Newcastle had further chances soon after the restart as Mark McGuinness did brilliantly to block Chris Wood's effort and then Sven Botman drilled wide from distance.
However, it was Wednesday who took the lead against the run of play in the 52nd minute.
Some superb play by George Byers opened the pitch up and, after working the ball out to the right, Windass was on hand to turn Dennis Adeniran's cross home from close range.
Replays suggested he was in an offside position but with VAR not in operation Wednesday were able to celebrate.
Newcastle should have instantly responded, but they found that Dawson was having a day to remember.
He produced a brilliant stop to keep out Anderson effort after a loose ball fell to the Newcastle midfielder eight yards out and then was alert to tip over Murphy's swerving shot from distance.
And those stops proved pivotal as the Owls doubled their lead in the 66th minute, with Windass grabbing his second.
He was played in by Michael Smith and made no mistake, firing past Martin Dubravka to send Hillsborough wild.
But Newcastle were able to find an instant response this time as it was their turn to be grateful for the lack of VAR.
Guimaraes was in a clear offside position as he poked home from close range after Dawson had kept out Wood's header from a corner.
The final 15 minutes were an onslaught, barring Windass' brilliant free-kick attempt, and Newcastle will feel they had enough chances to at least earn a replay.
Joe Willock had a shot blocked at the far post, but their golden opportunity came in the 82nd minute as Joelinton surged forward and teed up Wood who, with the goal gaping, inexplicably skied over to sum up Newcastle's night.