Further first half goals from Ezgjan Alioski, Jack Harrison and Rodrigo, and a brilliant strike in the second from Raphinha, gave Leeds their biggest top-flight victory since a 6-1 win at Charlton Athletic in April 2003 and confirmed the enormous task facing West Brom manager Sam Allardyce to keep his side up.
Leeds move up to 11th in the table with 23 points, while West Brom remain second-bottom on eight points from 16 games, five shy of safety.
The home side fell behind in the ninth minute as Sawyers put the ball into his own net. He was pressured 25 yards from his goal, swivelled and played a back-pass to where he expected goalkeeper Sam Johnstone to be.
But Johnstone had taken up a position to the left of his goal to give Sawyers an option for the pass, and the ball nestled into an empty net, a calamity from which the home side never recovered.
The match was billed as the defensive structure of Allardyce against the swagger and verve of Leeds under their manager Marcelo Bielsa, but it was proved no contest as the Baggies' wilted at the back and offered virtually nothing in attack.