Michael Turner had nodded the struggling Canaries in front at Emirates Stadium, but a controversial penalty allowed Mikel Arteta to level matters and swing momentum back in the Gunners' favour.
An own goal from Sebastien Bassong and a stoppage-time effort from Lukas Podolski saw the Gunners over the line, and they clamber above capital rivals Tottenham Hotspur and Chelsea in the top-flight standings.
The visitors started with some intent as on-loan forward Kei 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 Jack 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.
Following a positive spell by Norwich, Santi Cazorla tried his luck from range, but Mark Bunn was right behind it.
There was more purpose from Arsenal following the restart, if again without much penetration in the final third.
It was the visiting supporters, though, 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.
Arsene Wenger's response was immediate as he made a double change to bring on Theo Walcott and Podolski, replacing Wilshere and Gervinho.
It could, though, have got worse for the hosts when defender 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.
Arsenal were level with just five minutes left from the penalty spot. Following a corner, the assistant on the far side spotted a shirt pull by Kamara on Olivier 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.