Klich was given the nod over and above record signing Rodrigo and Tyler Roberts who are presently the other leading options in a no 10 role in addition to Pablo Hernandez who was missing altogether.
Like the Leeds display as a whole, Klich took a while to get going and was dispossessed on the edge of the box in the seventh minute by Ashley Westwood.
But the Pole soon began seeing plenty of the ball and in the tenth minute Klich calmly played in Patrick Bamford whose low shot from a tight angle was saved by Bailey Peacock-Farrell.
Klich had plenty of touches outside the box such as when linking up with Jack Harrison on the left flank and then when dinking in a cross in the 20th minute which was cleared.
Another cross seven minutes later was cleared before Klich provided a simple pass to Kalvin Phillips a few moments later to start another Whites attack.
His best moment yet arrived in the 35th minute when sending in Raphinha on the overlap down the right.
Raphinha provided a pull back for Gjanni Alioski who somehow sent his low shot out for a throw.
But Leeds were now well on top and brilliance from Klich gave the Whites the lead one minute before the break.
The Pole led another Whites break through the middle of the park and had several options ahead of him against a back-tracking Clarets defence.
But Klich opted to go for goal himself and unleashed a sublime curled attempt from the edge of the box that whizzed past Peacock-Farrell into the bottom right corner for his first goal since December.
The second half began in the same fashion as the first half with Klich doing a lot of chasing of the ball as Burnley looked to respond.
Defensive duties were called for such as when the Pole played a neat and calm pass as the Whites cleared a 56th-minute corner.
Whites head coach Marcelo Bielsa then made his first change in the 58th minute but Klich stayed on as Rodrigo instead replaced Patrick Bamford upfront.
Leeds doubled their advantage through Jack Harrison's clever finish two minutes later.
From Klich, more neat and short passes followed as Leeds camped themselves around the Clarets box and another cross from the Pole was cleared.
Klich was then free in the area as Leeds countered in numbers in the 75th minute but the move ended with a Dallas attempt being blocked and it proved Klich's last involvement.
The Pole was replaced by Roberts with 14 minutes left but it was the display of record signing Rodrigo from the bench that has likely left Bielsa with the biggest decision to make ahead of Tuesday night's clash at Southampton.
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Thank you Laura Collins