The home side went ahead in the 42nd minute at The Hawthorns when Claudio Yacob headed in Morgan Amalfitano’s cross.
But Wilshere hauled the Gunners back level in the 63rd minute when he thumped home Tomas Rosicky’s lay-off.
The draw leaves the north London side top of the Premier League table on goals scored heading into the international break.
“We’re happy with a point,” said Wilshere. “We’ll look back at this game at the end of the season and say it’s a good point – it’s a tough place to come.”
The England international continued: “It was a tough game for me. I’m not used to playing on the wing. I’m still getting used to it and learning but I’m happy to do it for the team.
“I’m part of a really good team here so it’s tough to get your place.
“I got a goal which was nice. I’ve always said that I wanted to score few more goals and hopefully when you score one a few more will come.
“I hope I’ll score more. Maybe in the past I wouldn’t have hit it as I did first-time, but it just fell to me and I thought I’d hit it. Luckily it went in.”
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