The Baggies took the lead in the 42nd minute at The Hawthorns when Claudio Yacob headed home Morgan Amalfitano’s cross.
But Wilshere hauled the Gunners back level in the 63rd minute when he thumped in Tomas Rosicky’s lay-off.
The England international had struggled to get into the game before the break but Wenger praised the 21-year-old for showing resilience against Steve Clarke’s men.
“He had trouble finding the rhythm of the game today but he has character – he doesn’t give up, he continues to play his game and in the end he got a great goal,” Wenger told Sky Sports.
“I believe as well, honestly, he was fouled a few times. He was pushed over and didn’t get the foul he deserved.”
Wenger, whose side moved back to the top of the Premier League table on goals scored, felt a point was a fair result against a battling West Brom.
“Over the 90 minutes it was a fair point for both sides,” said Wenger.
“Over the last half an hour I have some regrets because you could feel that we had opportunities to score a second goal.
“But you have to give credit to West Brom today, they confirmed the performance they showed last week [at Old Trafford].
“We again have showed character to come back into this game after being 1-0 down.”
Wenger’s side host Norwich City next after the forthcoming international break, and the Frenchman has been delighted by his side’s response to their opening-day defeat by Aston Villa.
He added: “It’s great for us because when you look at how we started in the first game, now we’ve just come out of two away games which are difficult for everybody. It’s a good position to start again after the break.”
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news