They eventually paid the price when the visitors pounced with eight minutes remaining to maintain their own unbeaten away record thanks to Eric Odhiambo’s goal.
Walter Smith’s side were looking for a tenth straight win, which would have been a first in 112 years, but it wasn’t to be on a frustrating afternoon.
While Inverness started brightly it was the home side that began to control the match and after a spell of pressure just before the 10 minute mark Edu made the break through.
He started the move, exchanging passes with McCulloch before bursting in to the box to make space for Steven Davis.
The Northern Ireland man picked him out and Edu tried to bring Miller in to the game, his first touch was poor but his second was sublime.
Where he missed with his right foot he made no mistake with his left, arrowing a shot in to the far corner of Ryan Esson’s net with unerring accuracy.
It was a brilliant strike and it sparked Rangers to up the tempo. Miller was next to try his luck after wriggling free in the middle of the park.
The prolific front man showed good strength to make the space but his shot from the edge of the area was easily gathered by Esson.
Miller was almost set up for an easier chance after 23 minutes when Kyle Lafferty showed great pace and determination on the left flank.
He beat Grant Munro the ball and then skipped past the Caley defender right on the touch line, but Miller’s effort from his cut back was deflected behind for a corner.
The visitors clearly weren’t happy with how little they were seeing of the ball and Richie Foran’s frustration bubbled over in the form of a terrible challenge on Edu.
The American needed lengthy treatment from physio Pip Yeates before he returned to the action for the remainder of the first period.
Rangers were always the more comfortable side but they couldn’t find another goal to increase their advantage before the interval.
Davie Weir and Lee McCulloch came close with headers while Caley defended well on the occasions when the home side did manage to produce a dangerous final ball.
Neither Edu nor Weiss would reappear for the second period with John Fleck making his first appearance of the season and Steven Naismith called back from the bench.
The visitors came out of the blocks quicker in the second period and forced some pressure on the home defence with Foran and Nick Ross both attempting to draw Terry Butcher’s side level.
The home side were then grateful to Allan McGregor for a brilliant piece of goalkeeping that kept them ahead.
Foran found space in the box and headed down and in to the ground and with the ball heading in under the bar McGregor showed brilliant agility to turn it over.
Two minutes later Esson almost matched that with a great finger tip stop from Fleck’s curling effort on the hour mark.
That signalled another period of pressure for the home side but they simply could find the killer touch that has been so often in evidence this season.
And when Inverness pounced on 82 minutes they made the most of their chance. Rooney won possession from Broadfoot and played in Foran.
He picked out Odhiambo’s run and the substitute did enough to bundle the ball over the line.
With so many late goals already this season the home support tried to roar their side over the finish line again, but Bougherra and Lafferty had the best of the chances with neither able to find the net.
The point was sufficient to keep the Light Blues top of the SPL but they will no doubt be disappointed that it wasn’t another win.