Cyriel Dessers and Ross McCausland were both on target in the second-half as the Light Blues began their defence of the trophy.
There were five changes from Tuesday evening’s match against Dynamo Kyiv as Kieran Dowell, Dujon Sterling, Vaclav Cerny, Rabbi Matondo and Danilo all came into the starting XI.
Ridvan, Leon Balogun and Scott Wright missed out on the matchday squad through injury, while Connor Barron, Ross McCausland and Cyriel Dessers were named on the bench for this match.
That meant Rangers started at Hampden with Jack Butland in goal and a back four of captain James Tavernier, John Souttar, Robin Propper and Jefte.
Mohamed Diomande remained in midfield, and he was joined there by Kieran Dowell and Dujon Sterling, while Danilo made his first competitive start of the season with support in the wide areas from Rabbi Matondo and Vaclav Cerny.
Rangers got the action underway at Hampden as Clement’s men looked to book their place in the quarter-finals of the competition.
Jefte and Rabbi Matondo linked-up down the left early on in this tie, with the former getting a shot away at goal but it was deflected wide for the first Rangers corner of the match.
Joshua Rae was forced into his first save with 15 minutes played and it again came down that left-hand side. This time Mohamed Diomande fed in Matondo, who drive forward before getting a shot away but the St Johnstone stopper got enough on it to push wide the strike.
St Johnstone had a good opportunity through Andre Raymond as he cut into the box and curled a shot towards goal, but John Souttar got a touch to turn it behind for a corner to the visitors.
At the other end, Souttar started another Rangers attack as he found Matondo with a long ball. The Welshman controlled the ball before feeding it to Jefte who had made an overlapping run. He whipped the ball into the box into the path of Danilo but the striker couldn’t keep his effort down as it drifted over the bar.
Rangers were pushing for a breakthrough and were denied by a double save from Joshua Rae with half an hour gone in this cup tie. The St Johnstone goalkeeper first managed to block a powerful strike from Mohamed Diomande, before he got a touch on a rebound shot from Danilo.
Jefte came close with a dipping strike from outside the box as the Light Blues continued their search for the opening goal in this tie, before Cerny fired over a fantastic opportunity after Danilo found him in space in the box but his right-footed strike was high of the target.
St Johnstone were denied by the post with five minutes of the first-half remaining, Matthew Smith found Adama Sidibeh at the front post for the Perth side, and his shot bounced off the woodwork before going out for a Rangers goal kick.
Clement was forced into a change before the break as centre-back Robin Propper had to be replaced through injury, with Ben Davies coming on in his place.
Rae pulled off another superb save deep into injury time as he once again denied Cerny from close range with a fantastic block with his feet.
At half-time, the Rangers boss turned to his bench again with a change in attack as Danilo was replaced by striker Cyriel Dessers.
Matondo was denied early in the second-half by Rae as his low drilled shot was blocked by the feet of the goalkeeper, before Cerny curled a shot just inches wide of the post after a he linked up well with James Tavernier on the right.
Rangers had the ball in the net with 58 minutes on the clock at Hampden as Rabbi Matondo played in Cyriel Dessers and the striker took a touch before sending a composed right-footed shot beyond Rae. The goal was initially disallowed by the referee for a foul in the build-up, but after a review by VAR, the strike stood to give Rangers the advantage in this cup tie.
There was another change for Gers just after the goal as Kieran Dowell was replaced by Connor Barron.
Matondo came close to adding a second as he linked-up with Cyriel Dessers inside the box but the winger’s shot was blocked by Rae, before Adama Sidibeh forced a save from Butland as the Rangers number one tipped his effort over the bar.
With just over 10 minutes of this match remaining Clement turned to his bench once again as Ross McCausland and Robbie Fraser were introduced in place of Vaclav Cerny and Rabbi Matondo.
Fraser was instantly involved in the action and delivered a lovely ball from the left, as he picked out desses but his shot was wide of the target. Fraser had a shot of his own shortly after as Rangers continued to push for that second goal but his strike from inside the box was held by Rae.
Sterling picked out McCausland with a lovely cross-field ball as the match approached the 90 minutes, but the winger could only find the side netting with a striike from a tight angle.
St Johnstone almost had a sight of goal as the ball broke for Adama Sidibeh inside the box, but Rangers were able to deal with the danger.
With two of the eight minutes of added time played, Rangers made the end to the game more comfortable with that all-important second goal. Ross McCausland combined with Barron and Dessers, before the Northern Irishman picked-up the return inside the box and made no mistake in sending the ball beyond Rae.
Rangers will find out their quarter-final opponents tomorrow afternoon, with the ties due to be played on the weekend of September 21/22.
RANGERS: Butland, Tavernier, Souttar, Propper (Davies, 44), Jefte, Diomande, Dowell (Barron, 62), Sterling, Cerny (McCausland, 83), Matondo (Fraser, 83), Danilo (Dessers, 46).
Subs: Kelly, King, McKinnon, Rice.
St Johnstone: Rae, Raymond, Cameron, Sanders, Neilson, MacPherson (Kirk, 75), Carey, Wright, Sidibeh, Smith, Kimpioka.
Subs: Sinclair, May, Clark, Smith, Sprangler, McPake, Olufunwa, Franczak.