Early in the second half, Zappacosta got around the back of City's defence, and was inches away from getting on the end of Kenedy's cross from the left, but Christoph Zimmermann just got there first.
"It's a bit frustrating".
Chelsea are now unbeaten in their last seven games in all competitions and visit Carrow Road looking to edge past a side who have recently emerged from a awful run of form to stay unbeaten in three games.
Norwich responded with their own attack and nearly took the lead from a 70th minute free-kick which was punched away by Caballero with Murphy's followup volley hitting a Blues player. "Offensively we were poor", he said.
"To be honest the Premier League title this year will go to Manchester City and the Champions League is not too easy to win, so the FA Cup is the biggest title and also a big opportunity for Chelsea". It's a pity because we wanted to win the game, instead we have to play another game'.
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"Norwich deserve to have another game".
Both teams will meet in a replay at Stamford Bridge to decide who goes through into the fourth round.
"For us it is a small problem as we have to play another game when the schedule is very busy". Having dominated the first half, the East Anglian side found themselves on the back foot after the interval, but Chelsea couldn't find a breakthrough. I think a draw is a fair result, but we are a little bit disappointed. They defended very well, showed they were well organised. It took until the 50th minute before the game's first shot on target - a weak effort from Michy Batshuayi, which was comfortably gathered by Angus Gunn.
Chelsea must now turn their attention to their EFL Cup semi-final first leg against Arsenal on Wednesday.