The Canaries launched a succession of raids down the right flank in the opening stages without finding the finish to test the Brighton keeper.
David Nielsen went close with a 25-yard drive that was inches away from the top left-hand corner of Michels Kuipers' goal.
In the 25th minute the Brighton keeper injured himself in clearing the ball and was replaced by 22-year-old Will Packham.
The Canaries continued to press forward but lacked quality with their crosses.
Gary Holt appeared to have set up Nielsen but Kerry Mayo nipped in to clear the ball off his toe.
Just before half time Bobby Zamora was booked after tangling with Malky Mackay.
Iwan Roberts replaced Zema Abbey after half time and he tested Packham with a header that the keeper juggled but kept out.
At the other end Robert Green could only palm away Richard Carpenter's teasing left-wing cross and Robbie Pethick stabbed the ball home from six yards to give the visitors a 48th minute lead.
It took Norwich eight minutes to draw level with Mulryne firing home a right-foot shot from 16 yards after Packham had done well to keep out a goal-bound shot from Roberts.
Green did well to beat away a fierce shot from Paul Brooker as the Seagulls made it clear they were not settling for a replay.
McVeigh put Norwich ahead in the 71st minute with a crisp 20-yard right-foot shot that curled inside Packham's right-hand upright.
The young keeper then saved a snapshot from Holt but was wrong-footed in the 81st minute when Mulryne's 25-yard free-kick took a deflection off Mayo to secure a 3-1 win.