In the second round of the Carabao Cup, a packed Elland Road was the scene of a Yorkshire derby as Elland Road was the venue for the game.
Luis Sinisterra scored from 25 yards and the hosts had a 2-0 lead.
The Tykes pulled one back through Mads Andersen's diving headbutt and then missed a spot-kick of their own.
Klich scored his second goal of the game with a curling shot.
Despite making nine changes to their starting line-up from the one that started on Sunday, they still took the lead at the end of the first half.
A brilliantly-struck effort from outside the penalty area opened the account for Sinisterra in his first start for the club.
Klich made it 2-0 by scoring the penalty after the impressive Sinisterra had been called for a foul.
The Tykes, who are 16th in League One, fought back after being knocked out of the first round of the playoffs.
The visitors had a chance to level when Styles was sent off for a foul on Forshaw, but he missed the penalty.
The home side regained their two-goal advantage in the 57th minute as Sinisterra's cross was not cleared and Klich scored.
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