The home side started the game well and threatened immediately through Santi Cazorla, but his effort from distance was comfortably saved by John Ruddy.
The Gunners took their time to click into gear but when they did, they took the lead with an excellent move.
Olivier Giroud and Wilshere combined and the Frenchman sent the 21-year-old through on goal with a lovely flick, and he made no mistake in firing a low finish past Ruddy.
Arsenal 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. He cut a pass back to Giroud, who fired a shot at goal but was denied by an excellent save from Ruddy.
Norwich almost found a leveller themselves as 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.