Sweden international Martin Olsson had opened the scoring on his first start for the Canaries before Anthony Pilkington doubled the lead just past the half-hour mark.
Elmander then got in on the act as he scored either side of a goal from Bury's Anton Forrester, with the League Two side getting another one back through William Edjenguele.
Summer signing Leroy Fer then scored his first goal for Norwich and Steven Whittaker added a sixth before Bury had the final say when Jessy Reindorf scored in stoppage-time.
Norwich started brightly but the opening goal did not come until Olsson struck just before the midway point of the first half.
Play was congested on the edge of the Bury penalty area but some patient build-up from the hosts allowed Elmander to slide a ball in to left-back Olsson, who tucked his chance away with aplomb.
The second followed shortly afterwards as Wes Hoolahan provided a slide-rule pass to Pilkington, who arrowed a shot past Rob Lainton for 2-0 and marked his first appearance of the season with a goal.
Kevin Blackwell's side came close to pulling one back against the run of play when Craig Jones struck a low shot that Mark Bunn did well to parry away from his near post.
That was the last meaningful action of the first half but a deflected Jones shot forced Bunn in to a fantastic save just seconds after the interval.
Those chances either side of the break were a warning to Norwich that their progression to round three was not yet secure, with Robert Snodgrass coming close to a third for the hosts.
That goal came in the 52nd minute with Elmander getting the goal his play until that point had fully deserved.
Snodgrass picked the ball up and slotted a perfect pass in to the path of the Galatasaray loanee, who made no mistake in thumping an effort past Lainton.
Scotland international Snodgrass was influential again moments later as his pass freed Pilkington down the left, with the ex-Huddersfield winger firing in to the outside of the post.
Elmander then squandered a great chance to grab his second of the game as he blazed over from close range when free inside the penalty area.
Four goals were then scored in a 12-minute period, with both teams pushing forward at every opportunity.
First, Bury finally managed to beat Bunn as Marlon Jackson headed on to the crossbar before Forrester was on hand to convert the rebound with the Norwich goalkeeper only able to get fingertips to the ball.
Norwich were unperturbed by Bury's goal and Elmander's second of the night came after some smart work from Snodgrass.
That goal did not signal the end of the visitors' attacking intent, however, and they scored their second of the night as Edjenguele headed home after Norwich had failed to clear a set-piece.
Remarkably, Norwich again responded and this time it was Holland international Fer who found the back of the net - firing in after Snodgrass' corner was only half-cleared.
That proved to be Fer's final contribution as he was replaced by Nathan Redmond with six minutes remaining, with fellow substitute Whittaker scoring his second goal of the season with a well-taken finish in the final minute.
There was still time for Bury to grab yet another goal and it was a fantastic effort from substitute Reindorf, who collected the ball a full 30 yards from goal before smashing an unstoppable effort past Bunn.