The Premier League champions improved in the second period after struggling in the first, but could not break down the impressive Championship hosts, who have earned a replay at Stamford Bridge.
Kenedy, who was given a starting role at left wing-back, had the game's first attempt in the third minute, but the Brazilian's low strike was always moving wide of the Norwich post.
A head injury for Norwich captain Ivo Pinto then stopped play in the eighth minute, and it took a bit of time for both teams to settle back into the third-round clash inside a noisy Carrow Road.
Josh Murphy was the player making things happen for the Canaries in the final third of the field, although the number 11 was lacking a final ball as the score stayed level after a fairly uneventful first 20 minutes of football.
A mistake from David Luiz, who was seen in a Chelsea shirt for the first time since November, almost presented Alex Pritchard with a chance in the 24th minute, but the ball bounced off the Norwich attacker before dropping wide.
Murphy had arguably Norwich's best chance of the first period when he broke into the Chelsea box 10 minutes before the break, but the 22-year-old fired his low effort well wide of the post.
Chelsea did not have a single attempt of note in the first 45 minutes, and the half time whistle blew with Norwich much the happier of the two teams after a positive half from the Championship outfit.
Danny Drinkwater then missed the Norwich post with an effort in the 53rd minute as Chelsea started to make their mark in the final third of the field.
Willian was next to come close for the visitors, who started the second period with much more purpose, as he brought a fine save from Gunn in the 56th minute.
Pritchard had Norwich's first real chance of the second period after working a yard of space inside the Chelsea box just past the hour, but the midfielder's effort was deflected over Willy Caballero's crossbar.
Murphy then had an opportunity after collecting a loose punch from Caballero, but the attacker saw his effort deflected wide of the post as the score remained goalless approaching the final 20 minutes.
Alvaro Morata came off the Chelsea bench in the 74th minute as the visitors looked to avoid a replay, and Luiz sent a long-range free kick just over the crossbar 12 minutes from time.
Norwich also introduced Nelson Oliveira late on, but neither team could make the breakthrough as the third-round match at Carrow Road finished level.
NORWICH (5-4-1): Gunn; Lewis, Pinto, Zimmermann, Hanley, Klose; Maddison, Tettey, Trybull, Pritchard (Wildschut 87'); Murphy (Oliveira 83')
CHELSEA (3-4-3): Caballero; Rudiger, Luiz, Cahill; Zappacosta, Drinkwater, Bakayoko, Kenedy (Musonda 78'); Pedro (Sterling 89'), Batshuayi (Morata 74'), Willian