Jack Wilshere and Mesut Ozil put Arsenal in control before Jonny Howson pulled one back for the visitors.
However, Arsene Wenger's men were not to be denied as an Aaron Ramsey strike and a second from Ozil sealed victory.German playmaker Ozil was included in the starting XI for Arsenal after recovering from a knee problem suffered on international duty, while Norwich were without striker Ricky van Wolfswinkel because of a toe injury.
Arsenal struggled to break through a determined Norwich defence early on, Santi Cazorla's weak effort their only opportunity of the opening exchanges.
However, after 18 minutes the deadlock was broken in magnificent fashion as Wilshere tapped home to finish off a wonderful move involving Cazorla, Wilshere and Olivier Giroud, whose improvised flick-on from the edge of the area sent the England international through on goal.
It took a further 18 minutes for Arsenal to create their next clear-cut chance, Norwich goalkeeper John Ruddy displaying excellent reflexes to prevent Giroud from doubling the lead from close range.
The hosts suffered a blow eight minutes before half-time as Mathieu Flamini was replaced by Ramsey following an earlier collision with Alexander Tettey.
And Arsenal's grip on the game looked to be in danger in the closing stages of the first half, with Robert Snodgrass and Leroy Fer both drawing strong saves from Wojciech Szczesny.
Norwich continued to press forward after the interval as Snodgrass and Martin Olsson tested Szczesny with long-range drives.
But their hopes of taking something from the game were severely dented when Ozil put Arsenal in control just before the hour mark.
Giroud was again the creator, his perfectly-placed cross finding Ozil, who rounded off a superb counterattack with a fine header into the bottom corner.
Chris Hughton's men did not lack ambition, though, and they found a way back into the game whenHowson drilled home a powerful low effort from 18-yards after 70 minutes when he capitalised on poor defensive work from Per Mertesacker.
Substitute Nicklas Bendtner then went close as Arsenal looked to finally kill the game off and victory was confirmed when Ramsey wrapped things up with an excellent goal.
The Wales international showed great agility to net his fifth Premier League goal of the season, slamming the ball past Ruddy from inside the six-yard box after weaving his way past two Norwich defenders.
Arsenal did not stop at three, though, and they put the gloss on a fine win with two minutes left whenOzil finished off another excellent move involving Tomas Rosicky and Ramsey.