The League Two side made life difficult for the much-changed Magpies in a fixture which was switched to Tyneside with the Wombles' pitch was unplayable following torrential rain last week.
The floodgates never quite opened here but Fabian Schär's penalty was enough to send United through, with the Magpies dominating possession but neither side creating much in the way of clear-cut chances.
Newcastle had the ball in the net in the 13th minute when Miguel Almirón - named in the starting XI as one of eight changes to the side which drew with Premier League champions Manchester City at the weekend - brought down a raking pass from Schär and squeezed home, but the flag was up against the Paraguayan.
Five minutes later, Joelinton, Sean Longstaff and Tino Livramento combined nicely for Longstaff to shoot, but his effort was saved by visiting goalkeeper Owen Goodman.
Schär was named captain for the night and his distribution from defence was a real asset for the Magpies. His pass to Harvey Barnes allowed in in-form winger to cut inside and let fly from the edge of the box, but his strike curled inches wide of the far post. Then from another Schär ball, Almirón half-volleyed over.
A crowd of 51,739 had assembled for the hastily-rearranged tie and they were nearly on their feet when Will Osula - making his full Newcastle debut - made space for a shot but fired off target.
The visitors' only real opening came in the 37th minute, when Martin Dúbravka was forced to make a brave block from Omar Bugiel and was hurt in the process. He carried on until half time, but gave way at the break for Odysseas Vlachodimos to make his bow for the club.
Before that came the decisive moment of the night. Almirón went down in the box under a challenge from Joe Pigott and referee Darren Bond eventually pointed to the spot, allowing Schär to step up and convert in first-half stoppage time.
United were in near complete control in the second half, with sub Bruno Guimarães seeing a shot easily fielded by Goodman before Riley Harbottle made a vital block to deny Joe Willock.
Almirón just failed to steer a deep delivery from Anthony Gordon in at the far post, then Osula spun onto Willock's pass in the box but drilled just wide.
Schär, Guimarães and Almirón all had opportunities too, and Osula was thwarted by Goodman. But while United couldn't add to their slender lead, that solitary goal was enough to see them through.
