The Gunners failed to extend their away record of eight wins on the road, but still returned to the top of the table on Sunday ahead of Liverpool on goals scored.
Claudio Yacob handed Albion the lead thanks to a fine header just before half-time, but Wilshere earned the visitors an away point with a 63rd-minute equaliser.
West Brom boss Steve Clarke adopted a three-pronged attack with Manchester United match-winner Saido Berahino handed his first league start alongside Stephane Sessegnon and former Gunner Nicolas Anelka.
And following a composed start from Albion, England Under-21 international Berahino tested Wojciech Szczesny with a heavily-deflected strike from 25 yards which the keeper did well to palm away.
The north Londoners gradually worked themselves into the game and began to dominate possession in Baggies’ territory.
Wilshere went close with a low left-foot drive which fizzed wide, while Boaz Myhill made an instinctive stop to deny Aaron Ramsey’s effort.
But three minutes before the break, Arsene Wenger’s side were left dumbfounded as they fell behind at a rocking Hawthorns.
Morgan Amalfitano’s delicious right-wing delivery was gobbled up at the near post by defensive midfield enforcer Jacob for his first goal for the club.
After the restart, Anelka came inches from doubling the hosts’ lead when he dragged his right-foot curler wide of the far post after being played on by Laurent Koscielny in the middle.
Just after the hour mark, Arsenal’s mounting pressure finally paid off as they restored parity with a well-worked move.
Olivier Giroud and substitute Tomas Rosicky combined to set up Wilshere, whose strike from the edge of the area took a slight deflection off Jonas Olsson, before flying past a wrong-footed Myhill.
It was the midfielder’s first top-flight goal for three years.
In a fast, frentic and flowing final 15 minutes, both teams has to settle for a share of the spoils, despite threatening to score the winning goal.
After the international break, the Gunners take on Norwich at Emirates Stadium before welcoming Borussia Dortmund in a mouthwatering Champions League Group F clash, while 12th-placed Albion face testing trips to Stoke and Liverpool.