Blues boss Conte, who made nine changes to his team, said: "I made many rotations and I'm very happy with Kenedy's performance".
Norwich came close to a famous victory in the first half when Alex Pritchard managed to intercept David Luiz's clearance, but he was unable to keep control of 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.
More fine play from Lewis allowed Murphy to tear through on goal in the 35th minute, beating Antonio Rudiger but getting his shot horribly wrong and firing across goal.
The major talking point of the half came when a poor touch from David Luiz almost allowed Alex Pritchard through on goal in the 23rd minute.
He was not at all fazed facing the likes of Brazilian defender David Luiz and England worldwide Gary Cahill.More news: Free flu vaccines offered at county health departments in TN
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".
While Willian tried to make something happen for Chelsea, he could not find a way past Norwich's defence, with Zimmermann particularly impressive in the back three.
That replay is likely to be played in about 10 days' time, stuck-in between the Leicester City and Brighton games, a week before going back to the Emirates for the second leg of the League Cup semifinal.
The last 32 will contest those ties over the weekend of January 26-28, with the winners to claim £90,000 in prize money. Unused subs: McGovern (GK), Husband, Vrancic, Watkins, Hoolahan.
The likes of Willy Caballero, Antonio Rudiger, Kenedy, Davide Zappacosta, Danny Drinkwater, Willian, Pedro and Michy Batshuayi are all pushing for recalls.