Right from the off, the visitors appeared unsettled by the gusting winds from Storm Ashley and an aggressive press from Kilmarnock.
Long balls to Kyle Vassell were proving fruitful for the hosts, who really should have taken an early lead.
Butland made great recovery save at the feet of Watkins, while Matty Kennedy failed to get a touch in front of goal as Corrie Ndaba delivered a fabulous cross.
Butland was beaten by a Robbie Deas header but the Kilmarnock defender had strayed offside from a free-kick and the first half ended not long after a shot from Vassell zipped over the top.
All Rangers could muster before the interval was a Vaclav Cerny shot that was beaten away by Robby McCrorie.
The former Ibrox goalkeeper was busier in the second half, tipping away an effort from Dessers, before the striker hit the crossbar from the resulting corner and Robin Propper knocked the rebound too high from close range.
McCrorie then saved well from Souttar after misjudging a cross, but Rangers could not build any sustained pressure.
Kilmarnock continued to offer a threat on the break and from set-pieces.
Kennedy miscued a couple of chances and Deas was guilty of a fresh-air swipe, but they were rewarded for their determination and endeavour when Watkins made the breakthrough.