Goals from Bilel Mohsni, Jon Daly and Lee McCulloch secured a 3-0 victory over the Borderers at Palmerston Park to set up the trip to Ochilview.
In the other semi-final, Championship side Raith Rovers host League Two outfit Annan at Stark's Park.
No date has been fixed for Rangers' clash with Martyn Corrigan's side while Annan's trip to Fife will take place on Sunday, October 13.
The final will be played on Sunday, April 6 next year but with the National Stadium at Hampden being readied for the Commonwealth Games a venue has yet to be decided.
The cost of hiring Celtic Park and subsequent policing costs has also ruled out the home of the Scottish champions as a possible host ground.
Rangers' home ground Ibrox has been raised as an option but Turnbull Hutton, a director at Raith Rovers, said his side will refuse to play there if they are paired with Ally McCoist's side.
"It is a sporting event and not an excuse to see how much money you can make out of the final," he told the Daily Record.
However, Henry McClelland, Turnbull's counterpart at Annan, takes the opposite view claiming: "I would see nothing wrong with maximising the earnings from the day."