CITY were made to rue first half missed chances as Arsenal eked out a 1-0 victory at the Emirates Stadium.
Heading into the match knowing that both Newcastle and Sunderland had picked up points earlier in the day, the Canaries were desperate to upset the odds against their hosts.
While Arsenal monopolised possession in the first half, City had the best chances - Nathan Redmond was thwarted twice by Petr Cech, and Gary O'Neil should have done better when presented with an opportunity close to the break.
Danny Welbeck's introduction early in the second half proved the turning point as he netted the game's only goal just three minutes after coming on, slamming home Olivier Giroud's lay-off.
With three matches left of their Premier League season, City are 19th in the table but face consecutive home games against Manchester United and Watford next to try and accrue the points needed for survival.
Alex Neil rung the changes from the 3-0 defeat to Sunderland a fortnight ago as five players started who didn't against the Black Cats.
Ivo Pinto, Russell Martin, Redmond, Wes Hoolahan and Cameron Jerome were all brought into the eleven as Neil implemented a different strategy at the Emirates.
Arsenal were unchanged from their 0-0 draw away to Sam Allardyce's team, but Santi Cazorla was on the bench - his first inclusion in a matchday squad since November.
Norwich showed promise in the final third in the early stages, with Hoolahan at the fulcrum of their attacking moves.
On seven minutes, Martin Olsson's deep cross found Redmond at the far post - and the winger's controlled half volley forced a save from Cech in the Gunners' goal.
Arsenal's first attempt saw the returning skipper Martin time a block to perfect to deny striker Olivier Giroud after tidy build-up play from Mesut Ozil and Aaron Ramsey.
The hosts were easily the side seeing most of the ball, with Neil's counter-attacking tactic utilising the pace of Jerome and Redmond the clear gameplan.
City had a concern on the half hour mark when Pinto required treatment following a heavy challenge by Laurent Koscielny, but the Portuguese full-back was eventually able to continue.
Up the other end, Koscielny was again involved in the action as his header flew over John Ruddy's bar after the France international met Ozil's corner at the near post.
Ruddy was yet to be tested, however, unlike his opposite number Cech who, three minutes before the break, kept out Redmond once again, this time after a volley from Hoolahan's cross.
There was still time for City to carve two more openings as O'Neil exchanged passes with Redmond before shanking his shot off target, and then Redmond himself dragged an effort wide from 18 yards.
It was an excellent finish to the half from the Canaries, but they didn't get the decisive goal and the whistle blew for the interval.
HALF-TIME: Arsenal 0-0 Norwich City
Norwich carried on the same vein in the first throes of the second half, with the home crowd growing frustrated at their team's failure to trouble City.
Arsene Wenger was then forced into a substitution on 51 minutes as Per Mertesacker hobbled off to be replaced by Gabriel, and then the long-serving manager brought on Welbeck in place of Alex Iwobi.
The former Manchester United man's impact was immediate - Welbeck waited for Giroud's knockdown in the box before sweeping the ball into the corner beyond Ruddy.
City needed to change tack to get back into the match, and Neil saw fit to bring on another striker in the shape of Dieumerci Mbokani. Hoolahan was withdrawn.
Redmond's industry down the right almost yielded an equaliser, but Gabriel defended superbly, timing his tackle to perfection to deny Mbokani a tap-in on the line.
As the tempo increased, Arsenal counter-attacked and fed Alexis Sanchez through the middle. The striker tried to outwit Ruddy with the outside of his right foot, but City's 'keeper read it all the way.
Ruddy then tipped a lasered Ozil shot from 25 yards over the bar, and City entered the final 15 minutes still chasing to overturn the one-goal deficit.
Steven Naismith came on for Robbie Brady, but Arsenal were still carrying the most threat and Ruddy had to be sharp to get a hand to Mohamed Elneny's shot on 82 minutes.
Three minutes before the end, City had a chance to test Cech with a direct free-kick. Redmond took it, but the wall did its job.
Neil brought on Matt Jarvis for Sebastien Bassong in one last throw of the dice, but Arsenal held on for the three points.
FULL-TIME: Arsenal 1-0 Norwich City
GOALS: Gunners: Danny Welbeck (59)
ARSENAL: Cech, Bellerin, Mertesacker (c) (Gabriel 51), Koscielny, Monreal, Elneny, Ramsey, Iwobi (Welbeck 56), Alexis (Coquelin 84), Ozil, Giroud. Subs not used: Ospina (GK), Wilshere, Cazorla, Walcott.
NORWICH CITY: Ruddy; Pinto, Martin (c), Bassong (Jarvis 89), Olsson; Redmond, Howson, O'Neil, Brady (Naismith 78); Hoolahan (Mbokani 68); Jerome. Subs not used: Rudd (GK), Whittaker, Dorrans, Mulumbu.
REFEREE: Mike Jones