Arsene Wenger's side had looked on course to move into second place on 29 points, behind City but ahead of Leicester on goal difference, when Mesut Ozil's 30th-minute goal underlined their early superiority.
But Lewis Grabban equalised against the run of play just before halftime with the eventual nervy draw for the pressurised Gunners leaving them fourth on 27, a point behind Manchester United.
Worse still for Wenger were three more injury blows to Alexis Sanchez, Santi Cazorla and Laurent Koscielny.
In an increasingly open-looking title race, with none of the fancied sides seeming to possess the quality to take control, Tottenham Hotspur also squandered an opportunity to bolster their challenge with a goalless draw at home to Chelsea, who dropped a clearly disgruntled Diego Costa.
Spurs have 25 points in fifth place but the Juergen Klopp juggernaut chugged on with the German manager's first league win at Anfield as Liverpool earned a laboured 1-0 victory over Swansea City with a James Milner penalty to go sixth on 23 points.
At Carrow Road, Wenger was left counting the cost of what he called a "frustrating" afternoon as Arsenal's "complete command" turned into an injury-riddled struggle.
"We have three injuries," Wenger told Sky Sports. "Cazorla played on one leg in the final part of the second half with a knee problem, Sanchez has a hamstring problem and Koscielny a hip problem.
"How bad it is, I don't know."
Wenger saw his "jaded" side miss the chance to cash in against Norwich, who are 16th in the table, following Saturday's draw between Leicester and Manchester United.
Ozil provided a delightful dinked finish to underline his current mastery -- a second goal to go with 11 assists in his last dozen Premier League appearances.
Yet poor defending from Gabriel let in Grabban and Wes Hoolahan almost put Norwich ahead moments later just before halftime.
CECH SAVE
Gunners goalkeeper Petr Cech had to make a brilliant second-half save at full stretch from his own defender Gabriel, and Wenger was left saluting Norwich's excellence as Arsenal struggled to hold on.
Champions Chelsea still proved toothless up front despite manager Jose Mourinho's axing of Costa at White Hart Lane.
The Spanish striker did not look a happy soul when, late on, he realised he was not going to be brought on as a substitute and, turning his back on the bench, tossed his warm-up bib behind him in disgust. It landed at Mourinho's feet.
Mourinho told BT Sport after the game that "there was not an issue, not at all" with Costa.
But even having produced what Mourinho felt was the best Chelsea performance of the season, a team with Eden Hazard as an unlikely target man still offered only intermittent threats in an unusually tepid London derby.
Spurs forced some decent work in the first half from Chelsea goalkeeper Asmir Begovic as they went a 13th straight league game without defeat, their best run since a 14-match streak in 1984-1985.
Hazard missed Chelsea's best chance with a wayward header and was also thwarted when his volley was brilliantly saved by Hugo Lloris as Mourinho's side, still unable to string together back-to-back league victories, struggle in 14th place on 15 points.
Klopp was left celebrating his sixth win in his last seven games in all competitions for Liverpool but may feel fortunate that Milner's 62nd-minute penalty was awarded after Swansea fullback Neil Taylor could do little about Jordon Ibe's cross striking his arm from close range.
West Ham United's sparkling start to the season continued to fade as Winston Reid's own goal just after the break earned West Bromwich Albion a 1-1 draw at Upton Park following Mauro Zarate's brilliant 17th-minute free kick.
(Reporting by Ian Chadband; Editing by Rob Hodgetts)
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