Jenna Fife was the hero of the day as she denied Celtic twice in the shootout after thwarting Saoirse Noonan from 12 yards previously in extra time.
It was a frenetic 90 minutes as Laura Berry, whose brace was added to by a Katie Wilkinson spot-kick in normal time, struck deep into stoppage time to force an additional 30 minutes with the scoreline level at 3-3.
The victory sets up a final with Glasgow City on Sunday, March 29 at Fir Park as the Gers chase a sensational four-peat in the competition.
The Head Coach named two changes to her side from last weekend’s creditable point away to the Petershill outfit, with Mia McAulay restored to the starting lineup following her return from injury.
Rangers began the brighter of the two teams as neat approach play on the right allowed Li Mengwen, also introduced to the team, to centre in search of Wilkinson but the cross was just slightly ahead of the prolific forward.
Jenna Fife was then equal to Maria McAneny’s header down the opposite end from a deep corner, before, just minutes later, Kim Shin Ji pinched possession and McAulay shot straight at Celtic goalkeeper Lisa Rodgers from Laura Berry’s pass.
The away side then broke the deadlock on the quarter of an hour mark when they tested the Gers early from a long ball, causing a collision between Jess Pegram and Morgan Cross that allowed Poppy Pritchard to break free.
It appeared under control as Rangers flooded bodies back, but they couldn’t quite clear their lines and the ball dropped kindly to McAneny on the edge of the penalty area who dispatched beyond Fife and into the bottom corner.
To their credit, the Gers wasted precious little time in being disappointed as they equalised six minutes later.
Berry was the architect as she received a direct pass from Shin Ji just inside the penalty area before turning smartly with Hannah Luke for company and the Celtic defender brought the Scotland youth international down.
Penalty duties were assumed by Wilkinson as she calmly slotted into the bottom corner to restore parity amid a frantic start in Cumbernauld.
And yet the hosts’ lead was short-lived as three minutes passed before Cross wriggled free inside her own half and found Pritchard in space. She had the freedom of the pitch to lift her head and slide in Emma Lawton running from deep who benefitted from a fortunate bounce off the onrushing Fife and stabbed the ball into an unguarded net.
The contest paused to take its breath for a period, but the away side extended their lead nine minutes before the interval to leave Rangers with a mountain to climb.
Cross received in space on the left-hand side, squaring up Mengwen before the full-back left a trailing leg in which the winger went down over for the awarding of a spot-kick.
Pritchard strode forward from 12 yards and dispatched in off the post to add a third for the visitors.
The next goal in contest was invariably going to be key, especially before half-time, and the Gers hauled themselves back into contention when Mengwen broke forward from Shin Ji’s pinpoint lofted pass before picking out Berry in the penalty area who stroked the ball beyond Rodgers.
The forward appeared on a one-woman mission to get the hosts back level as she tested her luck from distance shortly into the second period, but her shot was wayward and sailed over the top.
Opportunity then beckoned for McAulay just shy of the hour-mark, albeit from a difficult angle, when Shin Ji sent her racing in-behind on the break with a long pass, but her left-footed effort under pressure from Cross flashed wide of the target.
With 21 minutes remaining, Calliste Brookshire, an impressive performer last weekend against City, curled an inviting free-kick delivery into the penalty area which Lisa Rodgers flapped at and momentarily caused panic among the visitors.
Crichton reshuffled the attacking pack with the introduction of Camille Lafaix as Rangers sought an avenue at goal.
That nearly arrived when space opened up for Shin Ji from 25 yards out, but the midfielder drove her effort straight down the throat of Rodgers.
Time was beginning to run out on the hosts, with Quinty Sabajo introduced with six minutes to play, as they struggled to sustain much against the away side's backline.
Lafaix thought she had levelled in stoppage time when the ball broke invitingly at the far post from a Rodgers punch, but her effort deflected agonisingly wide of the far post.
No matter, though, as Rangers restored parity from the ensuing corner when Shin Ji's deep delivery was nodded against the upright by Wilkinson and Berry was on hand to knock home the rebound with one of the final actions of normal time.
An extra 30 minutes would decide the contest, with both sets of players out on their feet and requiring a second wind.
Berry was involved again midway inside the first period of extra time when she teed up Wilkinson on the edge of the penalty area, but the forward's effort was comfortable for Rodgers to collect.
Drama ensued moments later when Celtic were awarded a penalty - their second of the contest - when Leah Eddie was adjudged to have handled from Noonan's effort.
However, Fife produced heroics between the posts when she sprung down to her left to deny Noonan and keep the scoreline as it was.
The Gers goalkeeper then thwarted McAneny's dipping free-kick as the clock ticked towards half-time in extra time.
Nicola Docherty struck goalwards nine minutes from time which troubled Rodgers as she was forced to back pedal and tip over the crossbar at full stretch.
Rangers could have reigned supreme in the dying embers of extra time when opportunity opened up for Berry on the right-hand side, but she was denied by a sprawling save from Rodgers as a penalty shootout would determine the outcome.
The home side took first as Wilkinson converted, which was quickly answered by Kelly Clark of the away side.
Berry then rifled home, before Fife denied Claire Walsh to give the Gers the advantage.
The hosts' perfect penalties were continued through Shin Ji, with Lisa Robertson notching for Celtic to keep them within touching distance.
Brookshire confidently stroked home the fourth for Rangers, setting the stage for Fife to be the hero as she thwarted McAneny to send her side through to March's final.
RANGERS: Fife, Mengwen (Eddie, 57), Pegram, Rafferty (Hill, 70), Docherty, McLeary (Sabajo, 84), Shin Ji, McAulay (Lafaix, 70), Berry, Brookshire, Wilkinson
Subs not used: Kelly, Connolly-Jackson, Boyce
CELTIC: Rodgers, Walsh, Robertson, Gallacher (Westin, 104), Pritchard (Noonan, 78), Clark, Cross (Smith, 83), Luke, Lawton, Nakao, McAneny
Subs not used: Gay, Richardson, Ross, Dawson, McGoldrick
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