Fresh from their 4-0 rout of Falkirk in midweek, Celtic began the better of the two sides and almost took the lead with 10 minutes played.
Aiden McGeady did well to whip in a dangerous cross that Michael Fraser flapped at.
The ball then fell kindly to Scott Brown who promptly fired the ball goalwards only to see Phil McGuire clear his effort off the line.
Caley Thistle had rarely been seen as an attacking force but it took a wonderful save by former Inverness goalkeeper Mark Brown to keep out Marius Niculae's volley after Ian Black's initial effort had deflected into his path.
It was a miss that the hosts were left to rue as Celtic took a 24th minute lead.
Paul Hartley's cross from a free-kick found Jan Vennegoor of Hesselink's head and he made no mistake from six yards.
Less than 60 seconds later and Gordon Strachan's men doubled their advantage with the Dutch striker on target again.
McGeady was allowed too much time and space on the edge of the box to curl in a shot that was helped on its way into the corner of the net by Hesselink.
Celtic appeared to be cruising, but were made to pay for taking their foot of the gas three minutes before the interval.
Steven Pressley was adjudged to have brought down Niculae inside the box and John Rankin had enough power on his spot-kick to squeeze it under Brown's body.
There was no let-up in the goalmouth action following the interval and after Hesselink had struck the post, Caley Thistle equalised when David Proctor headed home Don Cowie's corner on 57 minutes.
Four minutes later and the home side's comeback was complete.
Richard Hastings' long ball found Cowie racing in between Pressley and Stephen McManus before stretching to prod the ball beyond Brown.
Celtic, however, were handed a lifeline on 67 minutes when Proctor clipped Scott McDonald's heels as the Australian raced in on goal, leaving referee Alan Freeland with no option but to give him his marching orders.
The visitors then piled on the pressure in search of an equaliser and only a spectacular save by Fraser from Massimo Donati's header late on prevented the points being shared.