And despite James Tavernier scoring from the penalty spot for the Light Blues on 89 minutes, the Staggies held on to inflict a painful defeat on Philippe Clement’s side in the Highlands.
With just a minute on the clock County, shooting towards the Union Bears in the first half, created an early opportunity and showed their intent.
Michee Efete found space to whip a dangerous right-foot cross to the back post and left-back Harmon scooped the ball up and over the bar.
It was a positive start from Don Cowie’s men and 10 minutes in a curling effort from striker Murray, at the end of a quick counterattack, went high and wide.
But for all of County’s efforts they went behind on 15 minutes, with Jack Baldwin guilty of putting the ball into his own net.
A teasing corner-kick from James Tavernier was flicked on by Fabio Silva and Balwin diverted past Ross Laidlaw, with the goalkeeper doing his best to stop the ball crossing the line but failing to do so.
With the Rangers fans in the ground initially hesitant to celebrate as it wasn’t clear if a goal had been scored, when referee Matthew McDermid pointed to the centre spot the bears in the Global Energy Stadium could finally enjoy the moment.
Six minutes after the opening goal, the Light Blues created another good chance. Todd Cantwell powered his way into the 18-yard box but his low, left-foot drive was well saved, with Laidlaw sticking out a leg to make an important stop.
At the other end of the pitch Jack Butland was called into action when Sims got into a shooting position and the Rangers number one had to dive low to his right to make an excellent stop to deny County’s winger.
At the half hour mark, however, Laidlaw was keeping his side in the game and stopping the visitors from doubling their advantage at this stage of the match.
The goalkeeper did brilliantly to get down to a clever header from Abdallah Sima which looked to dipping in at the back post. He then had to show lighting quick reactions to block a shot from Cyriel Dessers, who found acres of space in the penalty area and should have been more ruthless with his finish.
County went on to enjoy a good period of pressure, which resulted in the ball falling to Harmon but the defender lashed the ball over the bar from 10 yards when he had time to pick his spot.
With six minutes of the first half remaining a corner from Borna Barisic dropped to Sima and he came within inches of finding the bottom corner of the net.
Dessers then sent the ball over from an excellent cut-back cross from Barisic, which meant the score at half-time stayed at 1-0.
As with the first half, Ross County started the second on the front foot and two minutes into the restart they were back on level terms, with Murray beating Butland at the second attempt after the ball had been delivered in from the right flank.
On 50 minutes things were to get even worse for the Gers as the home side got their noses in front. Butland did well to make an initial save but the ball then broke for Harmon, who fired into the net from close range to stun the visitors.
Rangers tried to get back into the game quickly, with Dessers seeing a shot blocked and Laidlaw then doing well to stop a thunderbolt effort from Sima just before the hour mark.
With Dujon Sterling, Tom Lawrence and Rabbi Matondo now on from the bench, Sima had a goal disallowed for a foul on 68 minutes before Sims made it 3-1.
County’s number 11 received the ball from Eli King and confidently side-stepped Sterling before placing the ball beyond the helpless Butland.
With three minutes of the 90 remaining a penalty was awarded to Rangers via VAR after a handball from substitute Brandon Khela and James Tavernier stepped forward and calmly slotted home from 12 yards.
It was to be too late for the visitors, however, who – despite seven minutes of injury time added - will have to reflect on a disappointing and frustrating afternoon in Dingwall.
ROSS COUNTY: Laidlaw; Efete (Brown 78), Baldwin, Leak, Harmon; Sims (Khela 84), Randall, Loturi, King, Dhanda (White 84); Murray
Subs Not Used: Wickens, Borthwick-Jackson, Sheaf, Henderson, Brophy, Ayina
Goals: Murray 47, Harmon 50, Sims 69
Bookings: Baldwin, Efete, Loturi
RANGERS: Butland, Tavernier, Goldson, Souttar, Barisic (Sterling 54), Lundstram, Dowell (Lawrence 63), Cantwell, Silva (Matondo 54), Sima, Dessers (Roofe 73)
Subs Not Used: McCrorie, Balogun, King, Raskin, McCausland
Goals: Baldwin 15og, Tavernier 87
Bookings: Sima
Ref: Matthew MacDermid
Att: 6,382