Gunners held by Baggies
The Baggies, fresh from downing Manchester United last weekend, had ample opportunity to extend their lead after Claudio Yacob headed home a Morgan Amalfitano cross in the 42nd minute.
But it was Arsenal who ended the game in the hunt for the points after Wilshere, looking to replace reports of late night smoking with more positive headlines, scored with a deflected first-time shot from 20 yards.
While he has been criticised for lighting up in midweek, the England midfielder will have been happier with the spark he showed in a strong second-half showing.
Had Olivier Giroud been anywhere close to his best he would have settled the issue and put Arsenal two points clear by finishing one of two fine chances teed up by Mesut Ozil and Wilshere.
Albion began the match positively, Saido Berahino combining well with Stephane Sessegnon on his full league debut to free Liam Ridgewell on the left and earn an early corner.
Arsenal cleared the set-piece well and might have had a good break on their hands had the ball fallen to someone quicker than Giroud.
A deflection almost put the hosts ahead just before the quarter-hour mark, Berahino's shot changing course via Mathieu Flamini's leg and needing swift reactions from Wojciech Szczesny.
West Brom gradually took control of the midfield, Yacob and Youssuf Mulumbu proving an effective double act in the central area.
Wilshere was having less joy but came to the fore in 24th minute when he made a neat one-two with Giroud before sending a first-time shot a yard wide.
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