Two goals from Josh Windass after the break put the League One outfit in control, only for Bruno Guimaraes to quickly keep Newcastle in the game.
However, despite throwing numbers forward during the final quarter, Eddie Howe's side could not forced an equaliser as they suffered just their second defeat of the season in all competitions.
Despite making just three changes, Newcastle were matched all the way by their hosts during the opening 45 minutes, the main surprise being that neither side were able to break the deadlock.
Alexander Isak wasted Newcastle's best chance, seeing a header from point-blank range brilliantly kept out by Cameron Dawson, while Windass failed to convert having latched onto a poor Magpies back-pass.
However, he more than made up for it after the break, first putting the Owls into the lead with a smart finish after a superb flowing team move which had begun on the halfway line.
His second strike in the 66th minute was more of his own doing, latching onto a pass from Michael Smith before rifling the ball to Martin Dubravka's left and into the bottom corner.
Newcastle required just a matter of minutes to halve the deficit as Guimaraes converted from close range after Dawson could only parry a glancing Chris Wood header.
Sheffield Wednesday responded by almost netting a fairytale third through Windass, who was only denied a hat-trick when his 25-yard free kick hit the crossbar.
Back down the other end, Newcastle were soon piling on the pressure, and Wood wasted a golden opportunity from 14 yards when he miscued a shot high over the crossbar having been found in space.
Matt Ritchie also wasted a chance, shooting harmlessly over the crossbar from distance, and repeated last-ditch defending ensured that Sheffield Wednesday booked their place in the draw for the last 32.