Chelsea set the tone for their below-par performance in the first half when they regularly gave the ball away and failed to trouble Norwich goalkeeper Gunn.
The likes of Willy Caballero, Antonio Rudiger, Kenedy, Davide Zappacosta, Danny Drinkwater, Willian, Pedro and Michy Batshuayi are all pushing for recalls.
However, neither side showed the cutting edge needed to win and both clubs will go into the hat for the fourth round draw on Monday.
It is a fourth unbeaten game in a row for Daniel Farke's Canaries, who impressed at in defence but were shot shy in the final third, Caballero rarely tested despite a string of free-kicks in risky positions.
Steven Naismith has scored six goals in his last six matches against Chelsea in all competitions, including a hat-trick last time out in the Premier League in September 2015.
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Neither keeper was really tested as City forced a Stamford Bridge replay later this month.
Norwich made one change as Zimmermann replaced Oliveira.
Wastefulness in the final third was not something restricted to the hosts, with Chelsea lacking quality on their deliveries into the Norwich penalty area.
The Brazilian saw a couple of shots saved by Gunn and also picked out Drinkwater with a low cross from which he fired wide.
With 16 minutes remaining, Conte replaced the misfiring Batshuayi with Alvaro Morata, but even his first-choice striker was unable to provide a victor.
They confirm that Antonio Conte is the one who told David Luiz that he's staying, but the manager only did that because of Roman Abramovich's orders.
The result means yet another match for Chelsea in an already busy schedule.