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.
That ultimately proved to be Chelsea's last chance of the game.
That impressive run over the festive period has left Chelsea sitting third in the Premier League table - just one point behind Man Utd - but Antonio Conte is focussed on FA Cup matters this weekend.
A disappointing first half failed to produce a single shot on target, and, although Chelsea improved slightly after the break thanks to Willian, finding a goal proved too much.
Both teams will meet in a replay at Stamford Bridge to decide who goes through into the fourth round.
The Brazilian saw a couple of shots saved by Gunn and also picked out Drinkwater with a low cross from which he fired wide.
Chelsea made nine changes, with David Luiz making his first appearance since November 22 and Batshuayi, Caballero and Kenedy starting.
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Chelsea came out with more objective in the second half and had a chance on 50 minutes but Angus Gunn got down well to smother Michy Batshuayi's long-range shot.
Chelsea pressed late on but the closest they came to a victor was when Davide Zappacosta shot narrowly wide in injury-time.
The west Londoners have lifted the trophy four times in the last 11 seasons and they'll be confident of seeing-off Norwich in the third round of the competition this year.
Daniel Farke: "Perhaps they had some advantages but we still had our chances".
"Today it was a bigger chance to beat Chelsea than in the away game, but we will try everything", he said.
Conte admitted: "For sure there is a bit of disappointment because we have another match in a schedule that is so busy".
"Don't forget that Michy is a young player".