Their first league win at Rugby Park since October 2017 enabled Hibs to open a four-point lead over their opponents and move a point behind fifth-placed Livingston.
On the stroke of half-time, Hibs restored their lead through Adam Jackson and the visitors withstood second half pressure, with goalkeeper Ofir Marciano pulling off some impressive saves to secure a three precious points.
Hibs were first to threaten when Joe Newell had a goal-bound shot from Martin Boyle's cut-back blocked and Vykintas Slovka's effort on the rebound was blocked by Dario Del Fabro.
In blustery conditions at Rugby Park, both sides attempted to get the ball down and play and when the deadlock was broken it was through a moment of magic from Docherty in the 27th minute.
Newell did well to keep the ball in play and rolled possession back into the path of Docherty.
The midfielder, on loan from Rangers, cut inside from the left flank and curled a glorious right-footed shot from 20 yards out of the reach of Jan Koprivec and into the top corner.
Hibs lead lasted just three minutes as Burke drew Kilmarnock level.
Kilmarnock captain Gary Dicker displayed quick feet as he weaved his way into the Hibs penalty area and then passed to Niko Hamalainen, whose his low cross fell for Rory McKenzie.
McKenzie's drive from 12 yards was deflected into the path of Burke and the winger reacted well to send a diving header beyond Marciano.
Hibs went close to restoring their advantage through a quick free-kick from Newell as he played in Marc McNulty but Koprivec pulled off a fine save to deny the striker.
The visitors, however, did go in ahead at the interval as Jackson grabbed their second.
Dicker was penalised for a foul on Boyle and from the resultant free-kick, Newell swung in a cross which McNulty flicked on and Nicke Kabamba made a hash of clearing as it bounced off him into the path of Jackson for the defender to finish from close range.
Hibs were forced into a change just two minutes into the second half when Newell limped off and was replaced by Lewis Stevenson.
Kilmarnock pushed for an equaliser with Kabamba and Eamonn Brophy forcing Marciano into saves.
While at the other end, Koprivec denied Boyle when he was sent in on goal from a Scott Allan through-ball.
Marciano preserved Hibs' advantage when he tipped over Stephen O'Donnell's powerful drive and in stoppage-time, the goalkeeper pulled off a fingertip save to deny substitute Greg Kiltie.