But Miller restored parity as Gers turned in a much better second-period showing and at that point it was game on.
From there, the away side looked far more likely to win the match but they missed a glut of opportunities.
In particular, a missed penalty by Miller and another great chance deep in injury time which trickled just wide will be rued by manager Walter Smith.
Nevertheless, his men have moved a point further ahead of Celtic at the top of the league and they are now three clear with both teams having played 15 times.
With Sasa Papac returning from injury four days after Madjid Bougherra came back in Turkey, Smith was able to name his strongest team for some time.
Steven Whittaker moved up to the right side of midfield, with Vladimir Weiss on the left and Steven Naismith joining Miller up front in attack.
Inverness made the brighter start, with Russell Duncan having the first attempt at goal but firing well wide.
Allan McGregor then had to be alert to get to a long ball upfield just ahead of Adam Rooney as the hosts tried to get themselves an early advantage.
The away side were taking time to find any sort of rhythm and their defence was caught out again moments later, with Rooney played through on goal.
But McGregor showed why he is so highly rated as he came charging off his line to dive at the Irish forward’s feet and make a crucial save.
Assistant manager Ally McCoist was furious with the team’s lacklustre start and was quick to make it known from his position on the touchline.
His tirade prompted a response of sorts as Naismith headed just over the bar from Weiss’ smart reverse pass.
But it was the Highlanders who were doing most of the early running and Richard Foster had to get on the end of Hayes’ cross to ensure it wasn’t headed in.
The Light Blues slowly began to creep into the game but between a bobbly surface and some poor passing, they still weren’t anywhere near hitting top gear.
But then, out of nowhere, it was Rangers who very nearly snatched an unlikely lead against the run of play.
Steven Davis took a quick free kick to Naismith after he was fouled and he turned before feeding Miller.
With Ryan Esson coming out to close him down on the edge of the area, the forward got there first and dinked the ball over him.
But with the whole stadium holding its breath, it dropped the wrong side of the post and Caley Jags were off the hook.
While Gers had not heeded previous warnings, Inverness certainly did and a couple of minutes later, on the half hour mark, they went in front.
Hayes was the scorer, striking from almost 30 yards and picking his spot in McGregor’s top corner to give his side a lead it merited at that stage.
Going behind was nothing new to the Ibrox side, who had gone behind then won six times already in the SPL this season.
But a seventh victory in such circumstances was anything but assured and it would take something special against a notoriously stuffy team to come out on top.
It seemed there might be an opportunity to level from the penalty spot when Miller turned to shoot in the box and Ross Tokely appeared to block with his hands.
Yet despite furious protests from both the striker and Lee McCulloch, referee Calum Murray waved away claims.
McCulloch was then chopped to the ground by home skipper Richie Foran and a free kick was awarded, only for Weiss to blast it high and wide.
With half-time, the game was getting tetchy and Foran, Foster, Duncan and David Weir all saw yellow cards prior to the midway point.
The break allowed for a bit of restructuring in the away ranks and, sure enough, changes were made ahead of the resumption of play.
James Beattie was introduced for the wasteful Weiss and straight away Rangers went on the offensive, with Davis testing Esson from long range.
Whittaker then set up Miller, who rounded the keeper then beat him again before taking aim and sending the ball across goal and wide.
Beattie’s presence was making a difference while Davis was coming into the game and it took a deflection for his shot to roll out for a corner rather than towards the target.
And with 57 minutes gone, after their first real sustained period of pressure, the champions finally drew level.
Miller was the scorer with his 20th club goal of the season, stroking home beautifully from 18 yards out.
Suddenly the visitors were in the ascendancy and had Davis showed more composure, he might have picked out one of his team-mates in the centre.
The fact he didn’t nearly proved costly seconds later as Caley Thistle came forward and brought out a stunning double save from McGregor.
Rooney was first to test him but he blocked well and when Hayes headed the rebound back into the mix, a spectacular tip over the woodwork kept him out.
Back came Gers, with Miller’s threaded ball finding Whittaker and Beattie falling just short of poking it in.
The Englishman was then thwarted again as Stuart Duff clipped another cross away from him but Smith’s men were dominating and it seemed if anyone scored, it would be them.
Papac had a go with a free kick after Miller was brought down but there wasn’t enough dip on his left-footed effort and it was a yard too high.
Naismith was more guilty of a glaring miss when he broke the offside trap and with only Esson to beat, he somehow lifted above the bar.
It was rapidly turning into one of those days where it seemed Rangers wouldn’t get the result they wanted.
And when Beattie fell in the area and a penalty was given, Miller missed from the spot for the second time this term when he drilled right down the middle and Esson saved.
Time was running out and with just a couple of minutes of injury time to go, a frustrating draw was on the cards.
But there was one final chance to snatch the victory when Bougherra played low into the box for Miller – and when he turned just past, it summed up his day.
Rangers had to settle for their second SPL draw of the season at the final whistle, with both of them finishing 1-1 against Caley Jags.