Poland international Klich finished off a flowing United move to open the scoring in the 44th minute, bending his shot around centre back James Tarkowski and past the desperate dive of Burnley keeper Bailey Peacock-Farrell.
Harrison doubled United's 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 United's next two, 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 United sealed the points in style. Kalvin Phillips collected the ball on the edge of his own penalty area before playing a raking pass out to Harrison on the left.
Harrison found Rodrigo, who rounded Peacock-Farrell before slotting home. Rodrigo could have had a hat-trick as Leeds ran riot in the second half, but was denied twice by Peacock-Farrell.
"In the first half we insisted until we managed to score. We defended well," United manager Marcelo Bielsa said. "In the second half we didn't defend as well, but were efficient and managed to score."
Burnley started well, pressing high and disrupting Leeds' passing rhythm, but the visitors soon settled into their stride, and would have taken the lead in the 31st minute had Pascal Struijk not headed Phillips's cross wide.
The hosts offered precious little in attack, but United keeper Illan Meslier did well to keep out Matej Vydra's close-range shot in the 56th minute after the striker picked Struijk's pocket and ran through on goal.
Meslier repeated the trick against Johann Gudmundsson later in the half, denying the Iceland midfielder from close range with another good save to keep a rare clean sheet for United.
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. (Reuters)

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