Plymouth did not make it easy for the home side in spite of falling behind to Ched Evan' first goal for the Canaries early on, but the visitors only scored in the 89th minute after Darren Huckerby's penalty won the game for City two minutes earlier.
The Canaries kicked off attacking the River End goal and went ahead inside the third minute when Jamie Cureton released Evans who raced clear of the Plymouth defence and beat the advancing keeper low to his right with a right-foot shot from ten yards.
Peter Halmosi brought a diving save from David Marshall in the 18th minute and a minute later he was fouled by Jon Otsemobor who was booked for the offence.
It was half an hour before Norwich created another shooting chance but the keeper saved Cureton's shot with his feet. At the other end it was Marshall using his feet to keep out a shot from David Norris in a one-on-one situation.
The Norwich keeper had to be up on his feet to palm away a follow-up header from Barry Hayles.
Eleven minutes after the break Lee Croft replaced Matt Pattison and Evans was denied a second goal by the Plymouth keeper Romain Larrieu's diving save.
Cureton then wasted the best chance of the game when he blazed over from Croft's low right-wing cross with the goal at his mercy.
He was quickly replaced by veteran striker Dion Dublin and soon after Evans delivered a cross from the right, but Dublin headed over.
Darel Russell's shot was saved, but after he was fouled by Paul Connolly, Huckerby picked himself up to score from the spot his right-foot penalty sending the keeper the wrong way as the ball went to his left.
There was still time for Kristian Timar to touch in Halmosi's right-wing corner but not enough time for Plymouth to earn a point.