With Chelsea and Tottenham not in action this weekend because of the FA Cup semi-finals, the Gunners had the chance to put Champions League qualification back in their own hands.
However, the Canaries - who still need a couple of wins to secure top-flight status - produced a resolute battling display and took the lead on 56 minutes through Michael Turner's close-range header.
With the clock ticking, Arsenal were handed a way back into the match when the assistant referee spotted a shirt pull on Olivier Giroud by Kei Kamara. Mikel Arteta kept his cool from the spot to level, before Giroud's close-range effort turned the match around and substitute Lukas Podolski wrapped things up in stoppage time.
Arsenal host Everton here on Tuesday night and then head to Fulham before Spurs play against Manchester City next weekend, and so could open up a significant gap on their north-London rivals.
Chris Hughton's men, meanwhile, must now regroup for the visit of Reading and games with Stoke and Aston Villa, which will ultimately define the end to their season.
Arsenal midfielder Jack Wilshere returned from an ankle injury, and although fellow England international Theo Walcott was also fit again, but was named on the bench while captain Thomas Vermaelen returned to defence in place of suspended Per Mertesacker.
Norwich beat Arsenal at home in October which started a 10-match undefeated run, but have since stalled in 2013.
The visitors, though, started with some intent as on-loan forward Kamara offered support to captain Grant Holt and Robert Snodgrass in attack.
Arsenal, however, were soon on the offensive, as a flowing move down the left saw Gervinho cut the ball back across the face of goal.
On 20 minutes, a clever flick from Wilshere at the edge of the Norwich penalty area sent Aaron Ramsey clear, but the ball bounced just too far ahead of the Wales international, which allowed Mark Bunn to come off his line and collect.
Giroud was then unfortunate to see his backward header loop onto the top of the crossbar.
As the half-hour mark approached, Norwich were opened up again as Gervinho dashed onto a through ball and rounded Bunn as he dashed out of the area - but the Ivory Coast forward had been pushed too far wide on the right and his cutback from a tight angle rolled across the face of goal and behind.
Following a positive spell by Norwich, Santi Cazorla tried his luck from range, but Bunn was right behind it.
In stoppage-time, Giroud's bullet header from a corner was deflected behind by Scotland defender Steven Whittaker.
There was more purpose from Arsenal following the restart, if again without much penetration in the final third.
It was the visiting supporters who were in full voice just before the hour as Turner headed Norwich into the lead.
Kieran Gibbs, preferred to Nacho Monreal, tripped Kamara on the right edge of the Arsenal penalty area.
Snodgrass floated the free-kick towards the back post, where Turner rose unmarked to power a header into the bottom left corner after 56 minutes.
Wenger's response was immediate as he made a double change to bring on Walcott and Podolski, replacing Wilshere and Gervinho.
It could, though, have got worse for the hosts when defender Sebastien Bassong broke clear before releasing Russell Martin into the right side of the area, but the full-back's angled shot was straight at Lukasz Fabianski.
Podolski drilled in Giroud's clever chestdown some six yards out, but Bunn got just enough on the ball to tip it up onto the crossbar.
Arsenal were level with just two minutes left from the penalty spot.
Following a corner, the assistant on the far side spotted a shirt pull by Kamara on Giroud - which Norwich protested, and Bunn was shown a yellow card.
When calm was eventually restored, Arteta kept his nerve to guide the ball into the bottom-right corner just past Bunn's dive.
Arsenal turned things around when substitute Alex Oxlade-Chamberlain darted to the left and cut the ball back, where Giroud stabbed it home just ahead of Bassong's challenge.
After Jonny Howson's snapshot was saved by Fabianski, it finished 3-1 to flatter the Gunners when Podolski turned to drill home from 18 yards.