Valerien Ismael’s side missed a host of first-half chances before Callum Styles finally broke the deadlock shortly after the break.
Norwich looked nothing like the side taking the second tier by storm this season, and only threatened late on with Przemyslaw Placheta hitting the crossbar.
Styles also produced a heroic block to deny Placheta and help Barnsley hold onto a deserved victory, for which their reward will be another Oakwell clash in the next round against either Chelsea or Luton Town.
Story of the match
Norwich, seven points clear at the top of the Championship table, would still have been heavily fancied despite changing their entire midfield, but they were totally outplayed in a first half which saw them struggle to even progress into the opposition half.
Barnsley’s pressing and defensive organisation was superb, as was their creativity in attack, but the only thing lacking was the finishing touch as they somehow failed to take a lead into the break.
Tim Krul only just managed to scramble a swerving Cauley Woodrow strike early on, before the Dutch goalkeeper fielded two Romal Palmer testers and denied Michal Helik’s header from an Alex Mowatt cross.
Chances kept coming as Woodrow failed to beat Krul with a free header from a Dominik Frieser ball. Styles came close to knocking Callum Brittain’s cross in at the back post, and Norwich needed their stopper again when Frieser was allowed to fire from the middle of the box.
Two more big opportunities came before half-time, Victor Adeboyejo thrashing over the bar after an expansive move and Styles spotting an opening only to see his rasping drive pushed behind.
Some composure was badly needed in the second half, and it was provided by Styles 11 minutes after the restart. Opposite wing-back Brittain whipped in a cross from the right that fell all the way to the 20-year-old at the back post, and he shifted inside to fire into the far corner.
Striker Woodrow wasn’t having such luck and, having been denied a shot on goal by an excellent covering challenge from Norwich skipper Grant Hanley shortly before the opener, he had the chance to quickly double their lead only to scuff his attempted volley from a Frieser square.
At least he and his teammates were continuing to work chances; the visitors were not. Only in the final quarter of an hour did they begin to look anything like a threat, helped by the introduction of Emi Buendia from the bench.
His drive led to their first big chance, with Kieran Dowell letting his pass run for Placheta, but the winger failed to take the golden opportunity as he smacked the crossbar. That might have played on his mind a few minutes later when, played through on goal from a counter, he opted to pass to Buendia, whose goal-bound shot was heroically blocked by Styles.
Conor Chaplin was twice denied a clinching goal by Krul, ensuring a frantic, end-to-end conclusion. Norwich made a late push but their sole effort was a Jacob Sorensen shot fittingly kept out by match-winner Styles.
Reds imperious but one problem remains
This was without doubt one of the best performances, if not the greatest, of Ismael’s tenure at Barnsley so far.
His team were dominant for the vast majority of the game, pushing high and organising well from the start to deny Norwich any foothold in the game, and in contrast to recent matches showing the creativity and incisiveness required to create a host of chances as well.
But that perhaps only served to highlight one of this side’s weaknesses. Their forward players all had big chances – Woodrow and Frieser put guilt-edged opportunities straight at the goalkeeper, while Chaplin failed to convert three off the bench in the closing stages.
Their stranglehold on the game – Norwich didn’t register an official shot on target - meant that they were not made to pay, but they usually would – especially by a team like Chelsea, if they prove to be their next opponents.
Norwich fringe men fail to take chance
Daniel Farke was expected to make changes to his team and did so by switching the whole midfield and keeping the rest the same.
While nobody bar Krul could take much credit from this game, those who did come in failed badly to seize their chance to make an impression. Alex Tettey and Lukas Rupp meekly conceded midfield to their far more determined opponents, while Dowell and Onel Hernandez showed no creativity with the little forward ball they did have.
The other of the five, Placheta, had the chances to turn the match around, but after hitting the bar when he should have finished, he seemed to lack the confidence when through again minutes later and passed instead of pulling the trigger. This game will have done him no good at all.
Up front, Jordan Hugill cut a frustrated figure until being forced off by a hamstring injury in the second half. With Teemu Pukki and Adam Idah already on the sidelines, Norwich will hope for some reprieve on that front soon.
Man of the match: Callum Styles (Barnsley)
The young wing-back produced one of his best performances of an already hugely impressive season, capped by a clinical finish at one end and a miraculous block at the other.
Ismael said after the game that he had "returned from his deep sleep" after a few below-par performances, an assessment that can best be seen as a reflection of how high a standard Styles has set this year.