Despite creating numerous chances in front of goal the Hoops could not find the back of the net and Kilmarnock earned their first point at Celtic Park since their 1-1 draw in September 1998.
Celtic's start to the match should have signalled what was to follow with plenty of opportunities but a lack of a finishing touch.
Lee Naylor gave Alan Combe an early taste of the action with an effort that was well saved by the Killie man.
Jan Vennegoor of Hesselink should have scored on nine minutes, but when presented with an open goal, he screwed the ball across the target and out of play.
At the other end Colin Nish did well to hold off his marker before setting up Allan Johnston, but his effort deflected off a Celtic defender and fell straight to goalkeeper Mark Brown.
Rangers target Steven Naismith was not deterred by the boos coming from certain sections of the Celtic ground every time he touched the ball and came close to putting the visitors ahead with a 25-yard effort.
But after looking rattled by Killie's efforts on the counter-attack, Celtic took control of the game again and had a stack of unsuccessful chances to score. The visitors suffered a blow when James Fowler was stretchered off with a suspected broken ankle.
Aiden McGeady put his shot wide of goal while John Kennedy hooked over the crossbar in first-half injury time while Vennegoor of Hesselink had another effort cleared off the line by Simon Ford.
Celtic maintained their momentum at the start of the second half and within 60 seconds of the restart Paul Hartley had a header tipped over the crossbar by Combe.
Gordon Stratchan made an early change to his line-up bringing off Massimo Donati and Maciej Zurawski and replacing them with Chris Killen and Shunsuke Nakamura in the hope that the Japanese sensation would spark a goal.
Within seconds of coming on Nakamura was called into action with a 25-yard free-kick after Aiden McGeady has been bundled over but his effort was wide of the target.
But despite desperate efforts in the closing minutes, and with Kilmarnock down to ten men with substitute David Fernandez sidelined by a knock, Celtic still could not score and manager Strachan will have some concerns about his side at the start of a long season.