The hosts had taken the lead shortly before half-time on Sunday, courtesy of Claudio Yacob, before England midfielder Wilshere's strike earned a share of the spoils as the Gunners reclaimed top spot in the Premier League.
But the Frenchman was proud of Wilshere's contribution, despite the 21-year-old playing out on the wing for the majority of the 1-1 draw at The Hawthorns.
"He had trouble today to find the rhythm of the game," Gunners manager Wenger said on Sky Sports 1. "But he has character as well. He doesn't give up, he continues to play his game and got a great goal.
"I believe he was fouled a few times, pushed over and didn't get the fouls he deserved."
Wenger admitted a draw was probably a fair result, with both sides missing chances to win it.
"Over the whole game, I think it's a fair point. Over the last half-an-hour, I have some regrets, because we could feel we had the opportunities to score a second goal," he said.
"We again have shown character to come back into this game and in the second half being a goal down to equalise."
Meanwhile, Wilshere admitted he did not find it easy playing out of position, but is determined to do his best for the team.
"It was a tough game, coming here is never easy. We are happy with the point," he said.
"It was a tough game for me. I am not used to playing on the wing, I'm still learning it, but I'm happy to do it for the team. I got a goal which was nice, I've always said I want to score more goals."
The point lifted the Gunners back above Liverpool to the top of the standings, with the two sides level on points, heading into the international break.
"It's a good position to start again after the break," Wenger added.