Mowbray had called on Koumas to repeat his midweek goalscoring heroics for Wales when he scored twice for Liechtenstein.
It took Koumas just five minutes to do just that as he rammed home his first Albion goal of the season.
With Nathan Ellington on target two minutes later and Darren Carter adding a third in the 44th minute, Albion were assured of their first win in five games by half-time.
It was a vital success as a haul of one point from the previous four games had cast a cloud over their promotion aspirations.
Burnley never really recovered from the early blows delivered by Koumas and Ellington. They arrived at Albion boasting the best defensive record on the road in the Championship this season - having conceded just five goals on their travels.
But they looked anything but a defensive mean machine in the opening stages as they crashed to their second away defeat of the season - the first having come in their previous game at Cardiff City which was their first loss in eight matches.
To complete their misery, they had to play the final 12 minutes with ten men after former Albion loanee Frank Sinclair, who was making his 100th appearance for Burnley, was dismissed after collecting two yellow cards for challenges on Koumas.
Having hit the woodwork six times on the way to losing 1-0 to Norwich City in their previous game, Albion were more clinical against Burnley.
Koumas set them on their way when he collected a pass from Ellington and curled his shot beyond former Albion goalkeeper Brian Jensen. Burnley were still reeling from that setback when Albion doubled their advantage.
This time it was Diomansy Kamara who was the provider - cutting in from the right wing to set-up Ellington who converted from 20 yards.
To their credit, Burnley did not buckle and only a stunning save by Russell Hoult from Chris McCann's free-kick ensured Albion retained their two-goal advantage.
But when Michael Duff headed Sinclair's cross over the bar from six yards in 25th minute, the writing was on the wall for the Turf Moor outfit.
Kamara and Kevin Phillips both went close to adding to Albion's lead before Carter struck.
Once again, Ellington was to the fore, racing on to Phillips' pass and seeing his shot palmed out by Jensen. Unfortunately for Burnley, the loose ball flew straight to Carter who had the simple task of converting from two yards.