The Light Blues were taken to a quarter-final replay by the Second Division side with a 1-1 draw at Ibrox last week but there was never any danger of a shock at New Douglas Park.
Wide-man Fraser Aird gave Rangers the lead in the 18th minute when he volleyed in from a Calum Gallagher cross before striker Jon Daly headed in a second in the 57th minute from a Richard Foster delivery, but it was a competent display and no more from Ally McCoist's side.
The Ibrox club tied up the First Division title with a 3-0 home win over Airdrieonians last Wednesday and remain unbeaten in that competition, while they have a Ramsdens Cup final to look forward to against Raith Rovers at Easter Road next month.
However, if dreams of a trophy treble are to be realised they will have to be markedly better against the Taysiders albeit they will playing at their own stadium and backed by the majority of the support.
Gallagher, the 19-year-old forward who scored the second goal in a 2-0 home Scottish League One win over Dunfermline on Saturday after coming off the bench to make his debut, was handed his first start while midfielder Kyle Hutton was making his first start since last November after recovering from a broken leg.
As expected, the League One champions took control with Rovers' game plan quite clear, to sit back, frustrate, and try to hit on the break.
The Light Blues, consequently, had to be patient against a yellow and red wall which was set hard before them despite being built on the slippery artificial turf of Hamilton's ground, regarded as more suitable for the replay against the Govan club than Rovers' Cliftonhill.
There was no real sign of the breakthrough arriving when Gallagher sent in a cross from the right but Aird was sharp to the front post to cushion a volley past Cliftonhill keeper Neil Parry from less than six yards out.
In the 26th minute, as Albion opened up, Gers keeper Cammy Bell had to parry away a powerful shot from Rovers striker David Crawford with the Ibrox defence glad to get the ball to safety.
It was far from a hectic cup-tie, though.
In the 35th minute Albion's back-tracking wide-man Mark McGuigan deflected a goal-bound shot by Aird over the bar for a corner but again the Govan team could not capitalise and the first-half fizzled out.
Rangers lifted the tempo at the start of the second-half but only just and the narrow advantage meant there was no room for error.
In the 53rd minute Light Blues defender Bilel Mohsni headed an Aird corner from the right past the far post amid a sustained period of pressure.
It was dreary stuff but four minutes later the Gers effectively booked their semi-final place when Daly rose high inside the box to head a Foster cross past the helpless Parry.
There was noticeable relief among the large Rangers support as their team relaxed for the first time in the match and began to play.
Gallagher's right-footed shot drew a save from Parry before the keeper had to make a better block when Ross Dunlop's clearance cannoned off the onrushing Nicky Law leaving the shocked Rovers number one having to react quickly to knock it away.
The Gers midfielder had a chipped effort cleared away by Rovers defender Alan Reid in injury-time but the tie was over by then.
Rangers, however, will surely be tested more when they come up against Scottish Premiership opposition in their last-four tie.