The Light Blues had failed on their eight previous attempts to extend a winning run past two this year.
But after last week’s double triumph over Aberdeen, Danny Wilson’s winner seven minutes from the end sealed the long-awaited hat-trick of victories.
Craig Curren’s early opener put Owen Coyle’s team ahead but Alfredo Morelos first goal in 11 games put Rangers back on track.
The win will now be another boost to caretaker boss Graeme Murty’s chances of landing the Gers job permanently.
The under-20s boss was back in the dug-out after Derek McInnes stunned Rangers’ chiefs by knocking back their approach on Thursday.
The mood round Ibrox was understandably flat following that shock turn of events - but it took just 10 minutes for it to grow considerably darker.
Staggies right-back Jason Naismith burst down the right and swung in a cross over the head of Bruno Alves for Curran, who had drifted in behind his Portuguese marker. The striker still had work to do but confidently planted his header past Wes Foderingham.
Perhaps it was the shock of McInnes’ knock-back but Gers looked nothing like the side who had confidently out-passed Dons twice last week.
Too often they saw moves break down in their infancy. When they did sneak through they were met with stubborn resistance from County keeper Scott Fox.
Daniel Candeias and James Tavernier combined just after the quarter-hour mark but the latter’s low drive was palmed away.
Curran had a chance to punish Gers again but scuffed his shot. As the hosts raced forward, Jason Holt played a one-two with Declan John but, while Kenny Miller did well to pull off onto the midfielder’s cut-back, Fox was there with another stop.
It was another afternoon to forget from the anonymous Carlos Pena and Murty wasted no time in hooking him for Morelos at the break.
The Colombian had not netted for three months and it looked like his drought was set to continue as he snatched at his first couple of efforts.
But he finally got himself back on the scoresheet on 59 minutes. Miller’s interception set Gers off on the counter as Candeias was released into space and his cut-back was perfectly weighted for Morelos to stroke past Fox.
County should have scored again 10 minutes later as sub Ryan Dow drove into the box. With Davis Keillor-Dunn screaming to be played on, Dow went alone but his shot was weak.
But with the minutes ticking down Rangers grabbed the winner. Josh Windass’ header from Tavernier’s near-post free-kick delivery was palmed away by Fox but this time he sent it back into traffic, where Wilson was the first to pounce as he nodded home from close range.
Miller chipped onto the bar in stoppage time but there was a heart-in-mouth moment at the death as Naismith fired wide after a goalmouth scramble.