Two goals from Josh O’Connor either side of a Charlie Lindsay strike had the hosts ahead at half-time, but James Graham bundled home the equaliser with 15 minutes remaining and Robbie Ure scored two headers in four minutes to take the Light Blues through to the final against Hearts.
Hibs started strongly and were awarded a penalty after four minutes for handball, O’Connor stepping up and finding the bottom corner.
Lindsay squared the game on 20 minutes, beating Murray Johnson with a well-taken effort to level the score.
Ten minutes before the interval O’Connor scored his and Hibs’ second as he intercepted a slack pass out of defence and beat Jack McConnell with the aid of the post from 30 yards out.
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Rory Wilson might have drawn the visitors level again shortly before half-time but his effort drifted wide of the upright.
Kelsey Ewen headed just wide from a corner following the restart before Rangers brought on Graham and Kevin Ciubotaru on the hour mark and it was the former who hauled the Gers level when he bundled home from a corner following a stramash in the box.
Graham’s equaliser gave Rangers hope and with Hibs struggling to really test McConnell, it became virtually one-way traffic towards Johnson’s goal. With around eight minutes to play Ure’s header found the net and turned the tie on its head.
Ure then netted a second minutes later from Graham’s cross to seal Rangers’ place in the final against Hearts.