Alex Pritchard was unable to react quick enough to convert that half chance, while Josh Murphy failed to add to his eight goals in all competitions with a wild effort before the half hour.
The Premier League side, who reached the final of the FA Cup last season, created few chances of note after making nine changes to their starting line-up.
So, the two sides will have to do it all again at Stamford Bridge later this month, an additional game that Conte could do without given Chelsea are still competing in three other competitions.
Both teams will meet in a replay at Stamford Bridge to decide who goes through into the fourth round.
The Canaries sit 13th in the Championship table after winning just 3 of their last 13 games while they've been knocked out in the third round of the FA Cup for the last four seasons so they face a tough task today.
Norwich showed promising signs in the early exchanges without every worrying Caballero, only a misunderstanding between the goalkeeper and Luiz threatening to break the deadlock.
Chelsea manager Antonio Conte has admitted that Norwich City deserved their 0-0 draw in the third round of the FA Cup this evening. In the second half we tried to improve and created chances to score.
And Norwich boss Daniel Farke will be expecting the phone to ring with big clubs asking them how much they want for the wonder kid after he lit up the FA Cup.
Conte admitted that he was disappointed with the result and extra game to add to their already busy schedule, but also acknowledged that Norwich were worthy of the draw.
Chelsea manager Antonio Conte in his post-match reaction, expressed frustration of having to play a replay, adding that it was a hard game.
The German came into the side in place of Nelson Oliveira and helped keep Michy Batshuayi and then Alvaro Morata quiet in a solid display.
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