The relegation-threatened Canaries held their own for large spells in North London, but ultimately fell short to stay two points adrift of safety.
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The visitors, who dropped down to 19th in the table courtesy of results earlier in the day, created a big chance to break the deadlock after just six minutes when Martin Olsson's delivery from the left found Nathan Redmond, who took a touch before drawing an important save from Petr Cech.
That was the highlight of a promising start to the match for Norwich, and while clear attacking chances like that were few and far between, they did manage to keep Arsenal frustrated over the opening 45 minutes.
The mood inside the stadium was not improved when a number of fans protested against manager Arsene Wenger in the 12th minute, although sections of the ground did also voice their support for the long-serving Frenchman.
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The anti-Wenger brigade wouldn't have seen anything in the first half to change their minds, though, and it was Norwich who carried most of the attacking threat despite surrendering the majority of the ball.
Cameron Jerome brought a routine save from Cech shortly after the fan protest, but the bulk of the chances came at the end of the half, starting with another opening for Redmond that once again called for Cech to keep it out.
Gary O'Neil then fluffed his lines when well placed on the edge of the area before Redmond flashed another effort narrowly wide, with Arsenal only conjuring up headers from Laurent Koscielny and Per Mertesacker in response.
The second half continued in a similar vein and, after Mertesacker was forced off through injury, the crowd once again turned on Wenger when he replaced Alex Iwobi with Welbeck, instead of taking off Giroud.
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It proved to be a shrewd move from the Arsenal boss, however, as just three minutes later Giroud knocked the ball down to Welbeck, who fired home a deflected effort to give his side the lead with their first shot on target of the match.
That finally seemed to spark the home side into life, and the quiet Alexis Sanchez curled one a few yards wide from 25 yards having engineered space for himself to shoot.
It took a fine defensive challenge from Gabriel Paulista to deny Dieumerci Mbokani a clear chance at the other end with 20 minutes remaining, but Arsenal were back on the attack from the resulting corner as Mesut Ozil's sublime pass released Sanchez, who could only fire straight at John Ruddy.
Ozil then came close with a crisp volleyed effort of his own, but Norwich refused to give up on a point and had a half-chance with just over 10 minutes remaining when Sebastian Bassong couldn't react in time to steer a header on target.
Ruddy was forced into another save late on when he tipped Mohamed Elneny's effort over the top, but Norwich couldn't come up with anything at the other end of the field in the closing stages as they fell to a third straight defeat without scoring.
Arsenal, meanwhile, now sit third in the table ahead of both Manchester clubs playing on Sunday, with eight points separating them from fifth-placed United.