The visitors started strongly and finished even stronger, but Barnsley caretaker boss Simon Davey is still unbeaten in his four games in charge.
John Hartson, who came on as a substitute last week to score the winner against Derby, flicked on a header for Nigel Quashie but he drove his shot on the run at keeper Nick Colgan.
And when Jonathan Greening sent an astute ball to the right, Martin Albrechtsen tore to the byline but cut back his low cross behind Hartson and triggered a verbal assault from the former Wales international.
A minute later Kevin Phillips ran through on to a neat ball from Koumas but shot wide as an offside flag went up.
Barnsley, with two wins and a draw under Davey, might have edged in front after 19 minutes.
Leon Knight lifted a clever ball for Daniel Nardiello but Russell Hoult raced out to claw it away as far as Paul Hayes who smashed a 25-yarder which the keeper blocked and, as Nardiello followed up, the impressive Curtis Davies was there a fraction before.
After 23 minutes Nardiello produced a fierce effort from 25 yards but Hoult dived full-stretch to turn away the effort.
But on the half hour a quality move put West Brom in front. Hartson picked out Quashie and he lifted a perfectly flighted pass for Koumas who controlled smartly and then lashed a fierce shot past Colgan from just inside the area.
Five minutes later Barnsley were level. A Brian Howard wind-assisted corner hit the outside of the post and was played back into the danger area with Hoult strandedr. A shot from Nardiello was blocked on the line but Hayes was hovering to smash in the rebound from six yards.
And three minutes before the break the home side could have gone in front but Hayes headed just wide from Knight's clever chip.
Hartson came within a whisker of earning his side the lead 30 seconds after the restart when he used his full weight to smash a 25-yard shot that hit the post and Phillips followed up but the ball ran out off his leg.
Knight should have done better when Kyel Reid picked him out with a free-kick from the right but he headed wide.
And when Howard curled in another corner, Bobby Hassell was inches wide with his header.
Koumas came close to his second goal after 65 minutes when he hit a 25-yard screamer that Colgan had to tip over and then a Hartson shot hit Howard.
But Baggies boss Tony Mowbray went for broke when he brought on striker Nathan Ellington and went three up front for the final push for a win.
Ellington almost hit the winner five minutes from the end after Hassell's blunder but Colgan held on to his blistering shot at the second attempt.