The England international scored the equaliser at The Hawthorns as the Gunners came away with a point that keeps them top of the Premier League table.
Wenger, who had been critical of the 21-year-old during the week, was delighted that he ended a three-year dry spell in front of goal.
"He is an honest lad and when he's wrong he knows it," PA quotes Wenger as saying. "Today the players do not get away with anything - we got away with a bit more.
"They know that is the rule of the game and the rules of modern society and they have to deal with it, but what is more interesting for England and for Arsenal is that he shows the right response on the pitch.
"He had a difficult first half where he was fouled a lot but showed he has character and kept going. At least that [barren run] is done now."
When asked if Wilshere was still smoking, Wenger smirked and replied: "I don't know."