According to the BBC, the Baggies took the lead when an unmarked Claudio Yacob headed home Morgan Amalfitano's cross.
Nicolas Anelka missed two good chances to double the home side's lead shortly before Wilshere's deflected 20-yard shot levelled for Arsenal, the report said.
Olivier Giroud almost won it for the visitors but Boaz Myhill denied him, the report added.
Both sides continued to drive forward but struggled to break through, settling for a hard-earned point, which sends Arsenal into the two-week international break at the league's summit and lifts West Brom up to 12th, the report further said.
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