Austin’s 88th-minute penalty came just at the right time for Slaven Bilic’s side, who were coming heavy pressure at The Hawthorns before stretching their unbeaten run to seven games with their third straight win.
Hal Robson-Kanu’s fourth goal in six games gave Albion a 10th-minute lead but Steven Fletcher’s 58th-minute penalty appeared to salvage a deserved point for the Owls, who had Liam Palmer sent off in time added on in a game where both teams hit the woodwork.
In an exciting start, Robson-Kanu sent a twisting header looping over the bar in the first 30 seconds, from winger Matt Phillips’s cross.
Wednesday responded with a low cross from wideman Murphy, which the unmarked Fletcher was inches away from as he slid in at the far post.
The Owls continued on the offensive, forcing three corners in quick succession, and from the second of them, centre back Julian Borner’s snapshot was deflected wide.
Then it was West Brom’s turn to attack, and Filip Krovinovic curled wide from the edge of the box after a smart backheel by Robson-Kanu.
Albion took the lead when Matheus Pereira sent Robson-Kanu racing clean through with a reverse pass from halfway and the former Reading striker drew goalkeeper Keiren Westwood before calmly sidefooting home with his left foot.
After an improvement from the visitors at the start of the second half, they levelled through Fletcher.
The spot kick was awarded after poor play from Krovinovic, the midfielder allowing himself to be caught in possession by Lee, who he then appeared to trip as he lay on the ground.
Fletcher casually rolled the ball down the centre of the goal for his sixth goal of the season to end a six-match drought as goalkeeper Sam Johnstone dived to his right.
Robson-Kanu stabbed Phillips’s low centre just wide from point-blank range as Albion tried to hit back immediately.
But the pendulum had suddenly swung in Wednesday’s favour and they went desperately close to taking the lead.
Albion hit the woodwork on 87 when Austin’s glancing header hit the post following Diangana’s cross.
But Austin had not finished and his big moment came after Pereira was tripped by Westwood and the striker found the bottom corner from the spot.
Palmer saw red for lunging in on Diangana in the second minute of time added on.
There was still time for clean-through Diangana to be denied by Westwood as Albion finished strongly.