The Shakers, who sacked manager Chris Casper on Monday, played out of their skin for caretaker chief Chris Brass.
A goal in each half from Ben Futcher and Andy Bishop condemned the in-form Championship side to defeat.
Apart from the last five minutes when veteran striker Dion Dublin threw the Canaries a lifeline, the League Two outfit were rarely troubled.
Former York City boss Brass only received clearance to lead the team in the afternoon of the game.
Indeed, the match had been in doubt earlier in the day with referee Colin Webster passing the pitch fit after an inspection.
After a hesitant start Bury began to dominate proceedings.
Bishop missed a sitter after 17 minutes, but 60 seconds later giant defender Futcher shot home from eight yards after the Canaries failed to clear David Buchanan's cross.
Dublin came on at half-time and tried to grab the game by the scruff of the neck. The tall striker flicked the ball on for Jamie Cureton, but his effort was saved by keeper Jim Provett with one of three vital saves.
The East Anglians were made to rue their misses. Bishop, who put Bury in front at Carrow Road in the first encounter, netted his 12th of the season just after the hour.
Nicky Adams played in him with a superb pass and the former York City star did the rest with a left-foot finish across David Marshall.
Bury's task would have been easier had referee Webster sent off Darel Russell for a bad tackle on Adams instead of showing the midfielder just yellow.
Norwich got themselves back into it late on when substitute Lee Croft shot against a post before Dublin scrambled the ball home.
In stoppage time, Croft crossed and Doherty, who saved Norwich's skin in the first meeting, just couldn't connect at the far post to repeat his heroics.
Bury will now go to Southampton and after this display they shouldn't be written off.