With so much at stake, the tension was obvious as Rangers duo Mia Mcaulay and Nicola Docherty picked up early yellow cards.
Neither were deterred though, and both tried their luck from distance, although both efforts were easy for Noa Schumacher in the hibs goal.
The Hibs goalkeeper was called upon again as Mcaulay latched on to a brilliant through ball from Katie Wilkinson, but Schumacher produced a fine save.
While Hibs saw off that Rangers pressure, they created themselves. Eilidh Adams raced down the left, but couldn't find Ciara Grant, who was lurking unmarked.
The visitors also produced some lovely build up play as they looked to open the scoring, Grant this time getting a shot away, but it was straight at Jenna Fife.
Mcaulay was causing problems as she was allowed to run into the area unchallenged, but she should have done better with her finish.
With no goals in the first half, there was all to play for in the second 45, and both teams came out showing intent.
Hibs' Kirsten Reilly came close to breaking the deadlock, but her 25-yard strike crashed off the bar.
Rangers had plenty of possession, but weren't creating clear chances and Hibs were managing the game well.
Scott's side were full of desire, and when Kathleen McGovern worked tirelessly to keep the ball in the play, she then set up Taylor to convert a fine volley.
Rangers piled on the pressure looking for the equaliser.
Tessel Middag smashed a shot wide, McAulay blazed over, while a Katie Wilkinson header was weak. The goal would not come.
And Rangers misery was compounded as Laura Rafferty was showed a red card in injury time.
A poor afternoon for the Ibrox side, but euphoria for Hibs.