Jo Potter’s side twice led in the contest through goals from Mia McAulay and Lizzie Arnot before being pegged back.
And with the game evenly balanced as the two teams emerged for the second half, Rangers produced the pivotal moment in the tie seconds after Hibs ought to have taken the lead as Sarah Ewens beat the Hibs defence and finished with aplomb.
The away side then rubber stamped the victory with only minutes remaining when Rio Hardy netted twice, and Jane Ross got in on the act, to cap off a victorious afternoon for the Light Blues.
The triumph extends the Gers’ unbeaten campaign in all competitions, and they will now learn their last four opponents when the draw is made for the semi-finals tomorrow evening.
It was the home side who fashioned the first opportunity of the contest when Tegan Bowie drove forward on the left-hand side and crashed the crossbar with a speculative effort.
Rangers then took a firm grip of the opening proceedings and threatened the Hibs goal on nine minutes after Lizzie Arnot’s delivery was met at the far post by Brogan Hay whose control was immaculate but her finish off target.
The early pressure from the visitors paid off two minutes later as they broke the deadlock in the capital. Hay was the architect on this occasion as she skipped menacingly inside, laying the ball into the path of Mia McAulay whose driven effort from the edge of the penalty area arrowed into the bottom corner via a deflection from the Hibs defender.
However, Rangers’ joy was short-lived as they were pegged back against the run of play four minutes later. An inswinging corner from the left-hand side was met by Poppy Lawson from point-blank range who rose highest to head home.
The Gers never once started to lick their wounds as they went in search of an immediate response, edging their noses deservedly back in front in the 32nd minute. A sweeping move from Potter’s side caught Hibs on the hop, allowing goalscorer McAulay to hit the byline and lift her head to pick out Lizzie Arnot at the far post who reversed a lovely, left-footed finish into the bottom corner.
Despite their authority on the ball, however, the away side conceded for the second time in the half with seven minutes remaining. Indeed, it was a well-worked move from Hibs which cut the Light Blues open, with Kirsty Morrison afforded the space to pick out Shannon McGregor who dispatched a powerful finish beyond Vic Esson at her near post.
Rangers were unfortunate not to restore their advantage on the stroke of half-time when Sarah Ewens’ initial header was cleared only as far as Olivia McLoughlin, but the midfielder shot straight down the throat of the Hibs goalkeeper.
The away side began the second period where they left off in the first half, threatening the Hibs goal when Rachel Rowe aimed a free-kick from 20 yards under the wall that Nina Wilson was down quickly to smother.
Half-time prompted a shuffling of the pack from Potter as Rio Hardy replaced Arnot, with the former going close to finding the scoresheet just shy of the hour-mark when she was released beyond the Hibs defence by Nicola Docherty only for a last-ditch sliding challenge from Leah Eddie to thwart the forward.
The game then pivoted on a sequence where Rangers showed their ruthless efficiency. There was a heart-in-mouth moment when Esson was caught in possession deep inside her own defensive third, gifting possession to Morrison who looked to dink the goalkeepeer who was stranded off her goalline.
However, Esson was quick to recover to thwart the midfielder, with Nicola Docherty’s subsequent clearance catching Hibs fast asleep as the ball sailed over the defence and through to Sarah Ewens who advanced towards goal and dispatched a cool finish into the bottom corner.
Hibs were left shellshocked by the twist in fortunes, with Rangers looking to twist the knife when Rowe danced her way through the home side’s defence and looked to pick out Ewens positioned at the far post, but the goalscorer couldn’t quite bring the ball under control.
The midfielder was unfortunate not to find Rangers’ fourth – which would have surely eased all nerves – when she stole possession in the final third and had a rising effort tipped over the crossbar at full stretch by Wilson.
Any anxiety was quelled with four minutes remaining as Rangers were awarded a spot-kick after Jane Ross was brought down inside the penalty area. It was her strike partner, Hardy, who assumed penalty duties and the forward made no mistake when she planted her penalty into the bottom corner.
There was still time remaining for the Gers to add their fifth goal of the afternoon as Jane Ross showed tremendous opportunism to react first to a parried save inside the Hibs penalty area and knock the ball home.
The last action of the goal would, typical of Rangers’ second half display, produce a sixth for the away side as Hardy crept in at the far post to nod Rowe’s inch-perfect free-kick delivery beyond the Hibs goalkeeper.
HIBERNIAN: Wilson, Leishman, Hunter (Tweedie, 77), Baucom, Eddie, McGregor, Notley, Bowie (Christie, 63), Powell, Morrison, Lawson
Subs not used: Fraine, Erhabor, McAlone, Murray, Dorran-Barr
RANGERS: Esson, Docherty, Hill, Middag, Hay, Rowe, McAulay (McLeary, 77), Arnot (Hardy, 45), Ewens (Ross, 77), Cornet (Bance, 86), McLoughlin
Subs not used: Fife, Austin, Martinez