Klich's 85th minute strike turned a game, which Rotherham performed excellently in, around.
The Millers kept Leeds quiet for the opening 25 minutes and then smashed their way in front thanks to Semi Ajayi on 28 minutes.
Ajayi, being used in a defensive midifeld role, was allowed to roam forward from distance and following a smart passing move, he blasted an unstoppable strike into the top corner from 25 yards out.
Rotherham were stung when Leeds levelled against the run of play on 51 minutes with Klich prodding the ball over the line after indecisive defending from the home side.
There looked to have been a handball from the Polish midfielder before the goal but referee Tim Robinson was happy to award it.
Leeds were clearly energised by the goal and began to carve out more opportunities but without really testing Marek Rodak.
But the loanee goalkeeper was beaten in the 85th minute when Klich was found by Jack Harrison in the box and he squeezed an effort into the bottom corner.
Rotherham: Rodak, Jones, Raggett, Robertson, Mattock, Ajayi, Vaulks, Taylor (Yates 75), Towell (Crooks 85), Forde, Smith (Vassell 82).
Unused: Price, Vyner, Wood, Palmer.
Leeds: Casilla, Ayling, Cooper, Phillips, Alioski, Forshaw, Clarke (Roberts half time), Hernandez (Davis 90+5), Klich, Harrison (Shackleton 87), Roofe.
Unused: Peacock-Farrell, Pearce, Stevens, Halme.
Referee: Tim Robinson.
Attendance: 11,259 (2,317 from Leeds).