Ally McCoist's men fell at the first hurdle last season, losing 2-1 to Forfar after extra-time at Station Park.
The only two Premiership sides involved in the first round - promoted Dundee and Hamilton Academical - have home ties, against Peterhead and Arbroath respectively.
Scottish League One promotion hopefuls Falkirk must travel to Montrose and Morton have been drawn at home to Berwick Rangers.
Ramsdens Cup holders Raith Rovers begin their quest for more silverware against Forfar, while fellow Fifers Dunfermline host League Two side Annan Athletic.
All ties are due to be played on August 2, but the Queen's Park and Rangers tie will be rescheduled because of the Commonwealth Games; the game will be played at Airdrie's Excelsior stadium.
Scotland's European representatives - Celtic, Motherwell, League Cup holders Aberdeen and St. Johnstone - will enter the competition at the third round stage.
League Cup First Round Draw
Queen of the South v Elgin City
Brechin City v Dumbarton
Dunfermline Athletic v Annan Athletic
Queen's Park v Rangers
Morton v Berwick Rangers
Montrose v Falkirk
Raith Rovers v Forfar Athletic
Albion Rovers v Livingston
Hamilton Academical v Arbroath
East Stirlingshire v Ayr United
Dundee v Peterhead
Clyde v Cowdenbeath
Stranraer v East Fife
Airdrieonians v Stenhousemuir
Alloa Athletic v Stirling Albion