A stubborn Norwich had kept the drawbridge closed until a blur of Wilshere-inspired movement.
The midfielder exchanged passes with Kieran Gibbs, Santi Cazorla and Olivier Giroud before striking past Norwich keeper John Ruddy.
History was always against the visitors, who had scored in their last seven league games against Arsenal but had banked just four points during that run.
The Gunners lost enforcer Mathieu Flamini after 30 minutes following a clash of heads with Alex Tettey.
He was replaced by leading scorer Ramsey and with nothing to lose Norwich pushed forward but, sadly, without much conviction.
Arsenal were in control but that almost led to arrogance and Wenger will no doubt have a few words to say before Borussia Dortmund arrive on Tuesday for their Champions League game.
Wenger says it is a game that could define their season. They certainly go into it on a high and will need Ozil to continue in this rich vein of form.
The £42.4m man is pure class and scored Arsenal's second goal in the 58th minute, heading in Giroud's cross. But it was Wilshere's ability to stay calm when Norwich raised their game that was key.
Arsenal lost their shape for a spell, allowing Jonny Howson to score from a Martin Olsson cross on 70 minutes.
Norwich deserved a goal but Arsenal did not deserve to lose and Ramsey and Ozil's late goals ensured they did anything but.
"It is too early to be thinking about titles," said Wenger. "But if we maintain our attitude, don't get arrogant and keep the same belief, then who knows?"