Eamonn Brophy's first-half penalty was enough to send Kilmarnock back into third place in the Ladbrokes Premiership as Hibernian suffered a second consecutive defeat.
Brophy converted in the 32nd minute after David Gray was adjudged to have fouled Alex Bruce at a corner and the striker missed a glorious chance to double the lead as Killie dominated after the break.
There were some nervous moments towards the end, but the Ayrshire side held on in front of 7,484 fans at Rugby Park to claim pole position in the race for third and a guaranteed Europa League place.
Aberdeen's victory over Hearts on Friday night had put the Dons three points ahead of Killie, but Steve Clarke's men responded to take a six-goal advantage into the final day of the season.
Victory over Rangers will almost certainly assure Kilmarnock of a Europa League place while Aberdeen travel to Easter Road.
Fourth spot will only carry a European berth if Celtic beat Hearts in the William Hill Scottish Cup final.
The hosts started in determined mood and Bruce saw a long-range shot spilled as they applied some early pressure.
Hibs settled into the game and Daniel Bachmann got down well to save from Stephane Omeonga after the Belgian was set up by Fraser Murray.
But another spell of pressure told at the other end. Greg Taylor hit the bar with a cross and Ofir Marciano pushed wide Gary Dicker's header before the hosts won their penalty from the resulting corner. Gray and Bruce ended up on the floor as the ball sailed over towards the back post and Andrew Dallas pointed to the spot. Brophy stroked his penalty straight down the middle.
Hibs midfielder Stevie Mallan had a 22-yard shot saved before the break, but it was one-way traffic after the interval as Killie pushed for a second.
Brophy had a header saved amid a flurry of set-pieces and the striker squandered a golden opportunity 10 minutes into the half. Youssouf Mulumbu played him clean through with a first-time pass and Marciano hesitated to leave the course clear to goal, but Brophy shot just wide of the far post.
Marciano was quicker off his line to stop Taylor reaching Chris Burke's through ball and the Israeli soon saved from Brophy after Omeonga's loose pass.
Hibs head coach Paul Heckingbottom made three changes 20 minutes into the half with Thomas Agyepong, Daryl Horgan and Oli Shaw tasked with sparking a comeback.
There was little instant impact but Killie gradually drew deeper and the loss of Stuart Findlay to a head knock with five minutes remaining left them short of centre-backs with striker Kris Boyd coming off the bench and Dicker moving back into defence.
Agyepong soon threatened with a dipping long-range strike that Bachmann acrobatically tipped over and Killie saw out the final moments.
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