The hosts had the best of a hard-fought first half featuring few chances, the best of which saw Jamal Lewis set Murphy clear down the left in the 17th minute and the attacker's shot nearly caught Willy Caballero at his near post but the keeper claimed the ball.
Murphy went close again on 29 minutes as he cut in from the left only to see his shot go inches over the bar.
Here is how fans of the Canaries felt on their goalless draw.
Gunn's first real test came in the 56th minute, when he did well to keep out a deflected low shot from Willian at the near post.
With 10 minutes remaining, Murphy ghosted in from the right only to curl his strike over the bar.
"David Luiz is ready and will be in contention".
Asked if the Spain striker has been an effective replacement for Diego Costa, who has now joined Atletico Madrid, Conte added: "The stats are that Morata scored 12 goals this season, seven assists".
Chelsea have progressed from 20 of their last 21 FA Cup ties against non-Premier League sides, with the exception their 4-2 defeat to Bradford in January 2015.
The draw for the fourth round is scheduled to be made at around 7.10pm on Monday and is being televised on BBC Two and BT Sport, as well as on both of the broadcasters' websites.
Norwich made one change as Zimmermann replaced Oliveira.
The likes of Willy Caballero, Antonio Rudiger, Kenedy, Davide Zappacosta, Danny Drinkwater, Willian, Pedro and Michy Batshuayi are all pushing for recalls.