Emiliano Buendia was shown a straight red card in the 35th minute for striking Ashley Westwood with his arm after referee Kevin Friend watched a replay of the Argentinian's reaction to the Burnley midfielder's challenge.
Norwich were then reduced to nine men when Josip Drmic lunged in on Erik Pieters, with Chris Wood rubbing salt into the wound by scoring an overhead kick in the attack from the resulting free-kick.
Daniel Farke's team held firm for most of the second half but, despite being under no pressure, Ben Godfrey inexplicably touched Pieters' cross into the bottom-right corner to seal a miserable defeat.