French defender Mohsni grabbed a debut goal at the weekend against Arbroath and followed that up with another against hosts Queen of the South to give Ally McCoist's men an eighth-minute lead.
Yet that did not signal an early capitulation from the Dumfries side as their energy and determination made life difficult for the fallen former Scottish champions.
They had to wait until a quarter of an hour from the end to confirm McCoist's first last-four berth since taking up the Ibrox reins two years ago.
First Daly bulleted home a header just seconds after Chris Higgins was sent off for the home side after collecting a second yellow.
Then with six minutes left McCulloch made it seven goals in three games as he also nodded home.
The Ibrox club arrived in Dumfries with revenge at the forefront of their minds following their penalty shoot-out defeat to the Doonhamers at the same stage of last year's competition.
Yet Queens could have seen themselves level after 25 minutes had striker Derek Lyle not seen Mohsni block his strike with the Rangers goal empty following keeper Cammy Bell's mistake.
Midfielder Ian McShane also came close at the start of the second period but Rangers ground out the win.
They now face a decider with one of Annan Athletic, Raith Rovers or Stenhousemuir if they are to clinch their first final slot since the 2011 Scottish League Cup win over Celtic.
Rangers have barely been short of goals so far this season having racked up 23 in their five League One fixtures so far but it was their Championship hosts who came dangerously close to the opener.
Ricky Foster was nervous as he swiped McShane's early free-kick out for a corner and the same Queens midfielder put another dangerous delivery in from the resulting corner as he found Michael Paton at the back post. The striker's header looked net bound but thankfully for the Glasgow side, Arnold Peralta was on the line to clear.
And 60 seconds on from saving Rangers' skin, the Honduran international then played his part in his team's first goal.
His corner to the back post found Lee McCulloch, with the skipper firing a header back across goal for Mohsni to scramble over for his second strike in Rangers colours.
Home stopper Calum Antell then saved brilliantly from a long-range Peralta effort on the quarter hour before Derek Lyle shot over as the Light Blues hesitated inside their own box.
The early tempo was breathless and amid the frenzy, Rangers keeper Bell almost came a cropper after 25 minutes.
The former Kilmarnock stopper's heavy touch off a back pass let in Iain Russell to steal and set up Lyle with Bell sprawling on the floor. But again the Frenchman Mohsni was the visitors' hero as he blocked.
Yet the Doonhamers refused to relent as McShane skimmed the bar with a free-kick before Bell had to stretch to grasp Lyle's header.
The home side were quick out of the traps at the start of the second period but McShane's low shot from the right of the Rangers box squeezed wide.
Despite having the luxury of fielding as many of their summer signings as they pleased, Rangers could not settle. The fluency McCoist's men had shown in recent weeks was nowhere to be seen as they toiled to break down a resilient Dumfries defence.
Where chances had come often in the first half, the Glasgow side struggled to even gain territory in the final third.
But a rush of blood to the head by Higgins gave them impetus they need to complete the job.
Already on a first half booking for dissent, the Queens defender barged into Stevie Smith on the left hand touchline to collect a second. From the free-kick that followed, Nicky Law found Daly in between two defenders to power home the clincher.
McCulloch wrapped it up after 84 minutes as he smashed home another header from Robbie Crawford's corner, giving him a another goal to add to the back-to-back hat-tricks he netted against East Fife and Arbroath.