John Lundstram opened the scoring in the first-half, with a late Fabio Silva strike securing the win in the capital and ensuring Rangers’ place at Hampden next month.
Philippe Clement named an unchanged side from Thursday evening’s UEFA Europa League match away to Benfica which meant Rangers started at Easter Road with Jack Butland in goal and a back four of James Tavernier, Connor Goldson, John Souttar and Ridvan.
John Lundstram and Mohamed Diomande were joined in midfield by Tom Lawrence, while Dujon Sterling and Fabio Silva provided the support for Cyriel Dessers in attack.
Rabbi Matondo was back in the matchday squad as he returned from a hamstring injury and he was joined on the bench by Robby McCrorie, Kemar Roofe, Ben Davies, Leon Balogun, Borna Barisic, Nico Raskin, Ross McCausland and Cole McKinnon.
There was pressure from the first whistle for Rangers as they started strong in this cup tie with a series of corners in the opening two minutes. The first was flicked on by Fabio Silva deep into the box with Lundstram getting the final shot away but it was deflected wide.
Ridvan carved another opportunity for Rangers with 13 minutes on the clock as he made ground down the left before picking out Cyriel Dessers with an inviting cross into the box but the Nigerian forward’s first-time effort was just wide of the post.
There was almost an opening for Hibs shortly after as Myziane Maolida put pressure on Jack Butland on his line and came close to nicking the ball from the Rangers number one, but he recovered with Hibs winning a corner.
It was an open start at Easter Road, and Dessers had a good opportunity after he won the ball back in a dangerous position before driving into the box and cutting onto his left-foot to get a strike away but it was matched with a block from David Marshall.
With just over 20 minutes played, Rangers were awarded a penalty as Dujon Sterling was taken down in the box by Jordan Obita with referee Steven McLean pointing to the spot. After a VAR check, the penalty was confirmed with captain James Tavernier stepping-up.
Tavernier’s penalty was saved by David Marshall but John Lundstram was on hand to touch home the rebound and give Rangers the advantage in this cup tie.
Clement was forced into a change with just over half an hour played as Dujon Sterling had to be replaced after picking up an injury, with Ross McCausland coming on in his place.
There was a lengthy stoppage in the final five minutes of the first-half, with Martin Boyle receiving treatment for a head injury following a collision with John Souttar. The Hibs forward had to be stretchered off, with Elie Youan coming on in his place.
Nine minutes were added on at the end of the first-half, Diomande flashed a strike from outside the box just over the bar.
Into the second-half, John Lundstram did well to clear a Will Fish header as the home side enjoyed a spell early in the half in search of an equaliser in this cup tie.
James Tavernier forced a save from Marshall at the other end with a strong strike from distance, but the Hibs goalkeeper was able to get enough on it to touch it behind for a corner.
Emiliano Marcondes came close for Hibs with a free-kick in a dangerous position on the edge of the box as he curled the ball towards goal but Jack Butland was able to push clear.
There were more injury problems for Philippe Clement as Ross McCausland had to be replaced with an injury, as Rabbi Matondo came on for the winger. The manager also introduced Kemar Roofe in place of Cyriel Dessers.
Hibs were reduced to 10 men for the final 20 minutes at Easter Road as Jordan Obita received a second-yellow card for catching Matondo in the face with an outstretched arm as the pair raced down the touchline.
Just minutes later, there was another sending off for the home side with Nathan Moriah-Welsh shown a straight-red card for a lunging challenge on John Lundstram.
Rangers were looking to make their two-man advantage count to make a more comfortable end to this cup tie as Rabbi Matondo’s cross fell for Diomande in the box but his shot drifted just over the bar, before Fabio Silva had a shot turned behind for a corner in another attack.
Clement made the final two changes with just over 10 minutes of the match remaining as he introduced Cole McKinnon and Nico Raskin in place of Mohamed Diomande and Tom Lawrence.
That all-important second goal did come, and it was Fabio Silva with a fine finish to send Rangers one step closer to Hampden. John Lundstram picked-out the run of the on-loan Wolves forward who cut inside onto his right-foot and guided a low ball beyond Marshall.
In added time, Cole McKinnon had the ball in the back of the net as he reacted to the rebound from a James Tavernier free-kick but the youngster was denied a goal as the linesman raised his flag for offside.
It was a hard-fought victory for Clement’s men who will discover their semi-final opponents when the draw is made tomorrow evening.
RANGERS: Butland, Tavernier, Goldson, Souttar, Ridvan, Lundstram, Diomande (McKinnon, 79), Lawrence (Raskin, 79), Sterling (McCausland, 33, (Matondo, 65)), Silva, Dessers (Roofe, 65).
Subs: McCrorie, Davies, Balogun, Barisic.
Hibernian: Marshall, Fish, Boyle (Youan, 44), Newell, Cadden, Maolida (Tavares, 86), Marcondes (Mayenda, 86), Obita, Triantis (Levitt, 82), Moriah-Welsh, Bushiri.
Subs: Wollacott, Hanlon, Stevenson, Le Fondre, Whittaker.