Rangers kept probing and pressing with some patience but Killie’s rearguard held firm.
In the 27th minute, after Kilmarnock defender Clevid Dikomana tripped Itten 25 yards out, Ianis Hagi came close with a decent free-kick.
It was another set-piece which brought the goal after Tshibola conceded a free-kick 40 yards from goal.
The Killie defence failed to deal with Tavernier’s delivery into the box and when Tshibola hooked the ball clear it went only as far as Jack who returned it with a powerful volley high past Killie keeper Colin Doyle for his second goal of the season.
The home side might have doubled that lead moments later when Ryan Kent broke but his left-footed drive was tipped over by Doyle for a corner that came to nothing.