In fact, but for Izale McLeod’s header being cleared off the line – illegally according to the visitors – and Declan Rudd’s stunning save to deny Gavin Tomlin, the Red Devils would have been level with just minutes remaining.
But in the second minute of added time, Josh Murphy rifled in his second goal of the night to finally send the Canaries into the third round.
The League One visitors started promisingly, off the back of knocking Norwich’s East Anglian rivals Ipswich out of the first round.
But on 14 minutes, the hosts – much changed from their own derby win over Ipswich – notched the opening goal.
Scotland international Steven Whittaker cut inside but his low shot should have been held easily by Crawley goalkeeper Brian Jensen.
Instead, the ball spun free and Cameron Jerome scored his first goal for Norwich, following last week’s permanent switch from Stoke.
City appeared to have the game wrapped up when twin Jacob Murphy set Josh on his way through to slotting home the hosts’ second goal four minutes after the restart.
But Crawley were never out of it and another Norwich debutant, Carlos Cuellar, sliced in an own goal six minutes later to make life interesting for both sides.