The Ibrox manager saw his side secure their first semi-final slot under his charge as they beat Queen of the South 3-0 at Palmerston.
Rangers remain a division below Jim McIntyre's men following their liquidation and reformation in the bottom tier 12 months ago but McCoist has made a successful start to this season's promotion push with five wins out of five.
A first cup success would be an added bonus but he admits his keenness to first secure the League One crown is grounded in the fact failure will signal the end of his time in charge at Ibrox.
McCoist said: "It would be great to win the Ramsdens Cup but we want to win every cup we're in. We were gutted last season to be knocked out of it by the team we beat tonight.
"The one I want to win is the league because I won't be in a job if I don't get us promoted. But I want to win every competition we enter and we're still in the Ramsdens and Scottish Cup this season."
Rangers now face a semi-final tie against one of Annan, Raith and Stenhousemuir when the draw is made on Thursday.
They were ahead after just eight minutes as Moshni headed home after McCulloch fired the ball across goal from Arnold Peralta's corner.
Yet the Glasgow side failed to build on their lead and they Queens battled back into contention.
Moshni had to bravely block as Derek Lyle benefited from Cammy Bell's slip as he tried to clear a back pass after 25 minutes, while Ian McShane fired just wide just after the break.
But when home defender Craig Higgins - already booked for mouthing off at referee Calum Murray in the first half - picked up a second for a foul on Stevie Smith, it opened the door for Rangers to seal victory.
First Daly headed home from Nicky Law's free-kick with a quarter of an hour left while McCulloch added a late third from Robbie Crawford's header.
Queens boss McIntyre was delighted by his team's effort but admitted Higgins' rush of blood to the head cost them the chance of an upset.
He said: "The sending-off was pivotal. It was Higgins' man who scored from the free-kick.
"But I've got nothing but pride for the players because we really took the game to Rangers. They were ruthless and scored their first chance and we didn't have that side to us."
- Sports & Recreation
- Lee McCulloch
- Bilel Mohsni