Poland international Klich finished off a flowing Leeds move to open the scoring in the 44th minute, bending his shot around centre-back James Tarkowski and past the desperate dive of Burnley goalkeeper Bailey Peacock-Farrell.
Harrison doubled Leeds' advantage on the hour mark, getting a foot to Ezgjan Alioski's powerful drive through a crowd of bodies after a corner, to flick home past Peacock-Farrell.
Harrison turned provider for the next goal, scored by substitute Rodrigo, who chipped the ball over Peacock-Farrell from close range in the 77th minute.
The pair combined again three minutes later as Leeds sealed the points in style. Rodrigo could have had a hat trick as Leeds ran riot in the second half, but was denied twice by Peacock-Farrell.
Both sides are safe from relegation and the win kept Leeds in 10th place on 53 points, two behind ninth-placed Arsenal, while Burnley stay 15th on 39 points.
"In the first half we insisted until we managed to score. We defended well," Leeds manager Marcelo Bielsa said.
"In the second half we didn't defend as well, but were efficient and managed to score."