Wigan, who have managed just 25 goals in 29 games, dominated the first half but were out of luck in front of goal, with Emmerson Boyce and James McCarthy both hitting the bar among many fine chances.
They finally found the back of the net nine minutes after the break through James McArthur, but Scharner headed in a corner to level matters in the 65th minute, and a late flurry from the hosts came to nothing.
Franco Di Santo, in for the injured Hugo Rodallega, was the key figure as the Latics flew out of blocks, firing in three shots in quick succession inside the opening five minutes, the last of which forced Ben Foster to tip over the bar.
Not making the most of good spells in games has been the story of Wigan's season and from the resulting corner Gary Caldwell skewed a shot well wide from a great position, while a brilliant cross from Maynor Figueroa was met by a header from Boyce at the far post in the 16th minute that bounced off the top of the bar.
West Brom were offering little in attack, with Jerome Thomas' wayward shot their only real effort of the first half aside from a couple of Gareth McAuley headers from free-kicks.
The Baggies seemed to have weathered the storm as the game entered a scrappy phase, but Wigan came storming back at the end of the first half and really should have taken the lead.
McAuley did well to block a Di Santo header six minutes before the break and from the resulting corner the ball fell to McCarthy on the edge of the box, who fired in a thunderous shot that smashed against the bar.
When the ball finally came down the keeper reacted well to deny Victor Moses a tap-in, then moments later Jean Beausejour picked out Antolin Alcaraz in space at the far post, but he chose to try to find Shaun Maloney, making his first Premier League start for four years, who could not keep his balance with the goal gaping.
Wigan continued to probe at the start of the second half and in the 54th minute they finally got the goal they deserved as McArthur poked in his third of the season.
It was created by Moses, who showed good persistence on the right of the area after his initial cross had been blocked to beat Shorey and fire in a cross that Billy Jones could only direct to McArthur.
West Brom were almost level straight away as Thomas broke clear but Boyce did just enough to put him off and ensure he could not get his shot away.
Wigan probably should have had a second on the hour mark, Beausejour again showing his crossing prowess to pick out Alcaraz but the defender somehow failed to make any contact with the ball three yards out.
And the hosts were made to pay five minutes later when Brunt's corner was headed in at the back post by Scharner, who had been left unmarked.
Wigan fans were baying for a red card moments later when Youssouf Mulumbu reacted with fury to a challenge from McArthur and pushed him in the back, but referee Michael Oliver felt a yellow would suffice, with Boyce and Thomas also picking up bookings after becoming involved in the kerfuffle that followed.
Wigan knew what a great chance this was to claim that elusive home win and they continued to press forward.
Boyce had a clear sight of goal when he got the wrong side of Thomas but the winger did just enough to put him off and ensure his shot at Foster was weak.
Di Santo then saw another effort pushed away by Foster, although it looked to be going wide anyway, while substitute Keith Andrews was straight into the action with a shot just wide.
Wigan flooded forward in the final moments and came desperately close, first substitute Mohamed Diame miscuing a shot and hitting Di Santo when he should have scored and then Foster keeping out Albert Crusat's cross at the near post after it had been deflected by McAuley.
The point at least drew Wigan level with Wolves and QPR but they remain bottom of the table having played a game more than their rivals.