Given Antonio Conte's regular fixture complaints, he would not have wanted a replay, but Championship Norwich frustrated the Blues, who came closest when Willian's effort was turned round a post by Angus Gunn.
The last time City went beyond the third round was five seasons ago but there was no sign of an inferiority complex as Murphy exposed the Chelsea backline but fired harmlessly across the face of the goal and another chance went begging.
The Third Round replay will take place at Stamford Bridge on the week commencing the 15th of January; the official date will be confirmed as soon as the matches for television coverage have been selected.
Chelsea must now turn their attention to their EFL Cup semi-final first leg against Arsenal on Wednesday.
You always had the feeling Norwich had missed their opportunity when they dominated the first half without seriously troubling Willy Caballero.
The Brazilian saw a couple of shots saved by Gunn and also picked out Drinkwater with a low cross from which he fired wide.
It's claimed that David Luiz considered a move out of Chelsea in November after falling out with the manager, but in the end it was the club owner's wish that prevailed.
Norwich made one change as Zimmermann replaced Oliveira.
Chelsea pressed late on but the closest they came to a victor was when Davide Zappacosta shot narrowly wide in injury-time. A draw is a totally fair result.
"For sure there is a bit of disappointment because to have another match in a schedule so busy is not easy, is not simple, but we tried to win the game".
"We know it's an unbelievable task". When you play in this trophy you know it is very hard, but today the commitment of my players was very good.