The hosts made the brighter start of the two sides, with Wesley Hoolahan trying his luck with a shot after just four minutes.
Ten minutes later Chris Martin went for a spectacular overhead kick from inside the Scunthorpe penalty area, but failed to keep it on target.
Norwich eventually took the head in the 32nd minute when Simon Lappin found the target with a left-footed shot from 25 yards.
However, Scunthorpe restored parity just two minutes later, with Mark Duffy scoring from 30 yards.
The Canaries regained their one-goal advantage in the 56th minute when Wesley Hoolahan fired in from the spot.
Niall Canavan's foul on Hoolahan inside the Scunthorpe penalty area saw the referee point to the spot, and the midfielder made no mistake from 12 yards.
That goal eventually proved to be the winner, as Norwich advance to the third round of the competition.