Wildsmith got a hand to Trai Hume’s 36th-minute free-kick but should have prevented the left-back scoring the winner for a second successive match.
Sunderland’s ninth away win of the season consolidated fourth place and kept Albion – managed by former Black Cats head coach Tony Mowbray – seventh before the later afternoon kick-offs.
The hosts created chances galore but could not convert them, while Sunderland goalkeeper Anthony Patterson denied Kyle Bartley.
Darnell Furlong’s volley was blocked by Chris Mepham on the goalline after Patterson punched away Karlan Grant’s deflected cross, then midfielder Alex Mowatt ballooned a shot over the bar.
Sunderland eventually got out of their own half just after the mid-point of the first period when Romaine Mundle curled narrowly wide from the edge of the box.
Patrick Roberts went close for the visitors after cutting inside when his left-footed drive was deflected behind.
Sunderland were forced into a 34th-minute substitution when Mundle was replaced by Eliezer Mayenda – and soon after Mayenda won the free-kick that proved decisive.
Left-back Hume curled the ball into the top corner but Wildsmith will be disappointed not to have saved it.
It was the Northern Ireland international’s second goal in as many games after netting the only goal against Millwall on Wearside last week.
Albion claimed a penalty when Furlong fell after cutting inside his man but referee Tom Nield waved play on.
Jayson Molumby should have equalised from a free header from point-blank range but glanced wide from Mikey Johnston’s cross.
Grady Diangana then whistled a powerful shot inches over the bar from 25 yards as Albion pressed again.
The second half started similarly to the first, with plenty of possession from Albion before Sunderland created a chance.
Hume tried to repeat his goal with another curling effort but this time it finished the wrong side of the post.
Albion made a double substitution in the 62nd minute when Daryl Dike and Tom Fellows replaced Alex Mowatt and Diangana.
Molumby and Johnston – twice each – had long-range shots blocked then the latter blazed over as Albion tried to get back on level terms.
Patterson made his first serious save in the 81st minute when he tipped away Kyle Bartley’s header from Furlong’s cross, before substitute Tammer Bany hooked way over the bar.
Dike perhaps had Albion’s best chance but he sent a free header over the bar from inside the six-yard box from Fellows’ cross.
Wilson Isidor’s shot for Sunderland was deflected behind in a rare visitors’ attack.
And the Black Cats held on to record a third successive victory at The Hawthorns, while Albion have suffered consecutive defeats.
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