The home side started the game well and threatened immediately through Santi Cazorla but his effort from distance was comfortably saved by John Ruddy in the opening minutes of the game.
The Gunners took their time to click into gear but when they did, they took the lead with an excellent move.
The Gunners came close to extending their lead just before the break as Russell Martin gave the ball away in a dangerous area allowing Kieran Gibbs to raid inside the box and he cut a pass back to Giroud, who fired a shot at goal but he was denied by an excellent save from Ruddy.
Norwich almost found a leveller themselves when Leroy Fer surged forward through the midfield and powered an effort at goal from distance and Wojciech Szczesny was forced into making a good save.
After the break Szczesny was called into action again to deny efforts from Martin Olsson and Robert Snodgrass from outside the penalty area.
The Gunners took advantage of Norwich's lack of prowess in front of goal and doubled their lead with a slick move on the counter-attack, finished off by Mesut Ozil as he headed home from close range from Giroud's pinpoint cross into the area.
However, the Canaries were not going down without a fight and got back into the game in the 70th minute when Jonny Howson fired an effort from the edge of the penalty area into the bottom corner to get his side level.
Any hope of a comeback from the visitors was quickly ended by Aaron Ramsey as he scored a brilliant third for the Gunners. He beat two men with a lovely bit of skill inside the box and fired a shot past Ruddy to all but secure his side's win.
Ozil made sure of the victory with two minutes left in the game with Ramsey this time the architect. The Welshman powered into the penalty area and cut the ball back to the German international, who was left with a simple finish.