First-half injuries to central defenders Dedryck Boyata and Kristoffer Ajer disrupted the Hoops, who were stifled on the artificial surface by the hard-working home side.
Celtic’s impressive early start gave way to some struggle as the well-organised Ayrshire side closed out all avenues and when Ajer was unable to recover from a leg injury he was replaced in the 35th minute by winger Scott Sinclair, with left-back Kieran Tierney moving into central defence.
Some much-needed goalmouth action then ensued. Celtic midfielder Olivier Ntcham fired a 25-yard free-kick on to the roof of the Killie goal before Greg Taylor drove wide at the other end after Hoops keeper Dorus de Vries had spilled a Jones effort.
The Dutchman then saved an angled drive by Rugby Park skipper Kris Boyd.
Kilmarnock keeper Jamie MacDonald pulled a Sinclair shot out of the air in the 54th minute as Celtic upped the tempo after the interval.
The complexion of the game changed however, with 20 minutes remaining when the unmarked Mulumbu was picked out by Jones and from eight yards out guided the ball past De Vries for his first Killie goal since arriving in November.
The 31-year-old former Norwich City and West Brom player removed his shirt amid ecstatic celebrations for which he was booked by referee Kevin Clancy.
Rodgers threw on Odsonne Edouard for Eboue Kouassi and Celtic stormed forward in search of the leveller but Killie should have increased their lead.
Jones blasted over De Vries’ bar in a breakaway before Mulumbu had a drive blocked by the Parkhead keeper, who then saved another effort from Jones.
MacDonald saved a late free-kick from Ntcham and the determined home side held out the three added minutes.