St Mirren stunned Celtic 3-2 in the semi-finals last season and went on to beat Hearts by the same scoreline in the final in March to lift the trophy for the first time in their 136-year history.
The two sides met in pre-season, with Queen of the South coming out 4-2 winners, and boss Danny Lennon knows his team is going to have to do it the hard way if they want to retain the trophy.
He said: "We had a difficult run on our way to winning the trophy last season.
"But it's no different going into this year's competition as the holders than to any other season. We want to do well in all three competitions that we enter.
"A lot of people say that lightening won't strike twice, but it can. The trophy is something we want to hold on to.
"It's a tricky one for us. We had an opportunity during pre-season to go down there and play Queens.
"The positive is that we have had a wee flavour of their new astro pitch. It wasn't a pleasant night for us but Jimmy (McIntyre, Queen of the South manager) has inherited a very good squad and is now putting his own mark on it. So we know the challenge that faces us down there."
Forfar, shock winners over Rangers in the first round last weekend, have another chance to cause an upset after the draw paired them with Dundee at Dens Park.
Celtic, Motherwell, St Johnstone and Hibernian, who are or were in European competition, have byes to the third round along with Inverness, but the rest of the Scottish Premiership sides are involved at this stage.
Dundee United make the trip to Dumbarton, Kimarnock host Hamilton, Partick Thistle welcome Cowdenbeath to Firhill, Alloa face a daunting trip to Aberdeen and cash-strapped Hearts have a tricky tie at Raith Rovers.
Second round draw:
Morton v Montrose
Stranraer v Ross County
Dundee v Forfar Athletic
Dumbarton v Dundee United
Kilmarnock v Hamilton Academical
Airdrieonians v Livingston
Partick Thistle v Cowdenbeath
Queen of the South v St. Mirren
Aberdeen v Alloa Athletic
Raith Rovers v Heart of Midlothian
Falkirk v Dunfermline Athletic
Ties to be played Tuesday August 27 or Wednesday August 28