Amid the furore caused by Marcelo Bielsa's revelation he had spied on each of his Championship opponents training sessions this season, Leeds' promotion push had stumbled of late with three defeats in their previous four outings.
Another bad day seemed set to follow when they fell behind to Semi Ajayi's thunderbolt from 25 yards out after 28 minutes.
However, Klich struck early and late in the second period for his first goals since October.
In a further boost to Leeds' challenge to return to the Premier League after a 15-year absence, their closest challengers Norwich and Sheffield United played out a 2-2 draw at Carrow Road.
Norwich went ahead after 11 minutes when Onel Hernandez calmly finished between the legs of visiting goalkeeper Dean Henderson, but the Blades drew level on the stroke of half-time through Billy Sharp's penalty after Enda Stevens was fouled in the area.
Teemu Pukki restored the Canaries' lead when he flicked beyond Henderson before the hour, but Sharp bagged his 19th of the season when he headed home in the 79th minute.
Blackburn leapfrogged Hull and moved to within three points of the top six after a comprehensive 3-0 victory at Ewood Park.
Aston Villa are just a point behind Blackburn after the on-form Tammy Abraham's double helped them overcome rock-bottom Ipswich 2-1.
Abraham opened the scoring after just six minutes and then doubled their lead from the penalty spot just after the hour following Alan Judge's foul on John McGinn.
Freddie Sears struck after 76 minutes to set up a grandstand finish but Villa held on.
Martin O'Neill claimed his first win as Nottingham Forest manager as they saw off struggling Wigan 3-1.
Joe Lolley opened the scoring in the 19th minute and, although that was cancelled out by Josh Windass just after the half hour, Matty Cash restored Forest's lead just after the resumption of play.
Adlene Guedioura doubled Forest's advantage after 80 minutes before Latics substitute Joe Garner saw his late penalty saved by goalkeeper Costel Pantilimon.
Alan Browne and Brad Potts struck as Preston claimed a 2-0 win at Stoke, whose afternoon was compounded by Sam Clucas' penalty miss in added time.