Nico Raskin’s header was followed by a first Rangers goal for Bojan Miovski as Russell Martin’s men booked their place at Hampden in November.
From last weekend’s match, Derek Cornilus came in for his first start for the club while Nico Raskin also returned to the line-up.
Rangers started with Jack Butland in goal behind a back four of James Tavernier, John Souttar, Derek Cornelius and Jayden Meghoma.
Raskin returned to midfield to join Mohamed Diomande and Thelo Aasgaard in there, while Djeidi Gassama and Mikey Moore provided the support for striker Bojan Miovski.
There was an early opening for Nico Raskin after a good bit of build-up play but he flashed his effort wide of the target as Rangers dominated possession in the early stages.
Hibs had their first chance with 15 minutes on the clock as Josh Mulligan found Kieran Bowie with a ball through the Rangers defence, before he returned it to Martin Boyle, but his strike drifted over the bar.
Five minutes later, Boyle had the ball in the back of the net after he latched onto a long ball from Dylan Levitt. After a VAR check it was disallowed for a handball as it nicked Boyle’s hand in the build-up to the goal.
It was Rangers who took the lead with 42 minutes played as Nico Raskin came up with the crucial goal for the Light Blues on his return to the team. James Tavernier swung a perfect corner into the area, and the midfielder rose highest at the front post to guide a header beyond Sallinger.
Boyle threatened again for Hibs with a header of his own before the break, but it bounced just wide of Butland’s right-hand post.
It was a fantastic end to the first-half for Rangers, as they doubled their lead in injury time. Raskin was involved again as he sent a ball into the box from the right-hand side to find the run of Mikey Moore. The winger's shot deflected off of a Hibs defender and into the path of Bojan Miovski who made no mistake in getting off the mark with his first Rangers goal.
Early in the second-half, Djeidi Gassama clipped the bar with a superb strike on his right-foot after he cut inside to get a shot away from the edge of the box.
Rangers were growing in confidence and pushing for a third at Ibrox in a strong start to the second-half. Gassama intercepted a Jordan Obita ball back to his goalkeeper to lay it into the path of Miovski, but Grant Hanley rescued Hibs with a goal line clearance.
Miovski thought he had his second goal of the game as he tapped home from close range in the aftermath of another James Tavernier corner, but it was flagged offside.
Martin turned to his bench for the first time of the afternoon as he introduced Connor Barron into midfield in place of Mohamed Diomande, before Mikey Moore flashed a shot just wide as Rangers continued to push for a third goal at Ibrox.
There was a double change in attack for the Light Blues with just over 10 minutes remaining. Goalscorer Bojan Miovski and Mikey Moore were replaced by Youssef Chermiti and Oliver Antman, before Findlay Curtis and Nedim Bajrami were introduced for the final period.
Youssef Chermiti came close to his first goal for the club as he raced through on goal after a nice ball from Connor Barron, but his right-footed shot was blocked by the feet of the Hibernian goalkeeper. Antman was next to try and effort as he weaved his way into the Hibs box, but his low shot was just wide as the match finished 2-0.
Rangers will discover their semi-final opponents when the draw is made tomorrow afternoon.
RANGERS: Butland, Tavernier, Souttar, Cornelius, Meghoma, Raskin (Bajrami, 87), Diomande (Barron, 68), Aasgaard, Gassama (Curtis, 87), Moore (Antman, 77), Miovski (Chermiti, 77)
Subs: Kelly, Aarons, Djiga, Danilo.
Hibs: Sallinger, Hanley, Levitt (Barlaser, 59) , Bowie, Boyle (Molotnikov, 82), Chaiwa (C. Cadden, 82), McGrath (Klidje, 70), N. Cadden, Mulligan, Obita (Hoilett, 70), Bushiri.
Subs: Smith, O’Hora, Iredale, Barlaser, Campbell.