You always had the feeling Norwich had missed their opportunity when they dominated the first half without seriously troubling Willy Caballero.
Murphy went close again on 29 minutes as he cut in from the left only to see his shot go inches over the bar.
Chelsea welcome David Luiz back in to the squad after he recovered from a knee injury that's restricted him to just one appearance since October and the Brazilian worldwide is expected to start this evening.
Tiemoue Bakayoko fired wide, Willian forced Angus Gunn to push his shot around the post and the Brazil star then shot straight at the keeper.
With 10 minutes remaining, Murphy ghosted in from the right only to curl his strike over the bar.
Antonio Conte's response was to throw on £60million summer signing Alvaro Morata in place of the ineffective Michy Batshuayi up front, the Blues top scorer, with 12 goals to his names since signing from Real Madrid.
He was not at all fazed facing the likes of Brazilian defender David Luiz and England worldwide Gary Cahill.
While Willian tried to make something happen for Chelsea, he could not find a way past Norwich's defence, with Zimmermann particularly impressive in the back three.
Antonio Conte's men reached the final of the tournament last season and with the Premier League title looking out of reach, the Chelsea boss may target the FA Cup as a good source of domestic silverware.
The draw for the fourth round is scheduled to be made at around 7.10pm on Monday and is being televised on BBC Two and BT Sport, as well as on both of the broadcasters' websites.
The Canaries return to Championship action next Saturday when they travel to high-flying Bristol City. Unused subs: McGovern (GK), Husband, Vrancic, Watkins, Hoolahan.
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