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.
Wilshere admitted he did not find it easy playing out wide. “It was a tough game for me,” he said. “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.”
Asked about the smoking incident, he added: “At the end of the day, when it comes to the weekend you’ve got to do your job and today we’ve done it well.”
Wenger admitted a draw was probably a fair result. He said: “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. We have shown character to come back into this game.”
A deflection almost put the hosts ahead just before the quarter hour mark, Saido Berahino’s shot changing course via Mathieu Flamini’s leg and needing swift reactions by 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.
West Brom were aggrieved when Stephane Sessegnon was blocked as he looked to skip between Mikel Arteta and Kieran Gibbs and nothing was given, though contact did appear to be just outside the penalty area.
Aaron Ramsey, so prolific this season, went close when his low shot through a packed box was well saved by Boaz Myhill but West Brom were enjoying much more of the ball.
Their pressure paid three minutes from the break. Amalfitano’s initial corner was cleared but when Mulumbu produced a sublime piece of control and fed the ball straight back to him, the Frenchman produced a killer cross which Yacob met decisively with his head.
The second half began explosively, with wonderful chances at both ends inside the first three minutes.
The Baggies went first, Berahino galloping forward and making 30 yards before looking for Nicolas Anelka at the far post. The pass seemed perfect but Anelka made a mess of his finish and got the ball trapped between his feet.
Moments later, Arsenal advanced, with Giroud’s lay-off finding Wilshere ten yards out only for a wayward finish to spare West Brom.
The pace barely slowed before Anelka went even closer. Liam Ridgewell took advantage of a yawning gap on Arsenal’s right flank to play in the veteran, who moved towards goal and opened his body only to side-foot wide of the far post.
Arsenal had a penalty appeal for handball waved away when a Gibbs cross was blocked by Billy Jones, but the full-back appeared to do everything he could to withdraw his arm.
They were level after 63 minutes, with Wilshere the scorer. Ozil and Giroud were the instigators, with the latter rolling the ball to substitute Tomas Rosicky on the edge of the box.
His one-touch pass sat up for Wilshere, who did not hesitate in unleashing a left-footed shot from 20 yards. A deflection off Jonas Olsson turned a decent looking effort into an unstoppable one.
Arsenal were buoyed and probably should have taken the lead when a cute chip by Ozil presented Giroud with a delicious chance to volley from ten yards. He snatched at the shot, though, leaving Ozil to shake his head.
Giroud’s struggles continued when a rejuvenated Wilshere picked him out with a glorious ball over the top. This time the combination of a heavy first touch and a brave smothering save by Myhill denied him. Jones saw a 25-yarder well saved by Szczesny, but it appeared Arsenal’s game to win.