A double from Abdallah Sima, a first Rangers goal for Ridvan and a late Ryan Jack strike ensured Beale’s men their spot in November’s semi-finals at Hampden.
From Sunday’s league victory over Motherwell, there were four changes to the line-up as Ridvan, Ryan Jack, Abdallah Sima and Kemar Roofe were all named in the starting XI.
Rabbi Matondo was out of the matchday squad having picked up a knee injury at the weekend, while Borna Barisic, Scott Wright and Cyriel Dessers are named on the bench.
That meant Rangers started this cup tie with Jack Butland in goal and a back four of James Tavernier, Connor Goldson, Ben Davies and Ridvan.
John Lundstram and Jose Cifuentes were joined in midfield by Ryan Jack, while Sam Lammers and Abdallah Sima provided the support for striker Kemar Roofe.
It was Rangers who got the action underway at Ibrox as they looked to progress to the last four of the competition. An early corner, swung into the box by James Tavernier, saw centre-back Ben Davies get a touch to send it goalward but it was just over the target from the defender.
The Light Blues were looking lively, and from a series of balls into the box, it broke for Tavernier inside the area with the skipper releasing a powerful strike on target but Shamal George managed to get a strong hand to tip it over the bar and out for a corner.
The early pressure paid off for Rangers as they took the lead in this cup tie inside the opening 10 minutes. Ridvan picked out Abdallah Sima with a nice pass down the left, with the on-loan Brighton man winning the ball before cutting inside and curling a lovely right-footed finish into the back of the net.
Sima had another good opportunity to double the scoring shortly after as Jose CIfuentes found the forward’s run in the box but his touch was blocked by George.
With just over 35 minutes played, Livingston had their first real sight of goal as James Penrice broke forward with pace before finding Stephen Kelly on the edge of the box. The ex-Ranger had space and tried to curl an effort beyond Jack Butland but his effort drifted wide of the left-hand post.
There was one change for Rangers at the break as Cyriel Dessers was introduced in place of Kemar Roofe in attack for Beale’s men.
Sima was causing problems for Livingston all evening, and was denied a second goal of this match by Shamal George with a strong save from a close range shot.
John Lundstram was next to have a go at goal as Rangers looked to increase their advantage in this quarter-final, but his strike from distance was just over the bar.
David Martindale’s side had a rare opportunity from a corner that was swung into the area with Luiyi De-Lucas rising to get a good header towards goal but it was just wide of the post from the Livingston defender.
However, with just over 65 minutes on the clock, it was Rangers who managed to get that all-important second goal at Ibrox and it was a fantastic moment for left-back Ridvan. The 22-year-old picked the ball up in the middle of the pitch and went on a superb solo run through the Livingston defence driving into the box before finishing low beyond George for his first Rangers goal.
With just over 10 minutes of the match remaining, Sima launched another Rangers attack as he picked out the advancing run of James Tavernier with a lovely ball. Tavernier may have had an opportunity to shoot, but opted to cross the ball into the path of Sam Lammers who was unable to keep his effort down as it drifted just over the bar.
Rangers were in control at Ibrox and were three goals ahead in the final stages of the match. A long ball over the top released Abdallah Sima and from a tight angle the forward managed to send a shot beyond George.
With four minutes added on at Ibrox, Rangers made it four on the evening as Ryan Jack got in on the scoring with a neat finish from the edge of the box as he slotted low beyond George.
RANGERS: Butland, Tavernier, Goldson, Davies (Souttar, 88), Ridvan (Barisic, 72), Lundstram (Sterling, 88), Cifuentes, Jack, Lammers, Sima (Wright, 88), Roofe (Dessers, 46)
Subs: McCrorie, Balogun, Rice, McKinnon.
Livingston: George, Obileye, Stephen Kelly (Pittman, 69), Mackay (Sangare, 76), Brandon, Holt, Nouble, De-Lucas, Sean Kelly (Shinnie, 69), Guthrie (Anderson, 76), Penrice (Lloyd, 76).
Subs: Hamilton, Parkes, Bradley, Lawal.