Heart of Midlothian saw off Cove Rangers 3-0 at Tynecastle while Inverness CT defeated Peterhead 2-0 in the Highland capital.
Both matchday two games are on Wednesday, with Arbroath up against East Fife at Gayfield and Dundee United playing Elgin City at Tannadice.
Dundee thumped Brora Rangers 4-0 at the Kilmac Stadium and Forfar Athletic beat Montrose 5-4 on penalties after a 0-0 draw at Station Park.
Raith Rovers notched up a 4-0 victory over Brechin City at Stark's Park while Cowdenbeath earned a 1-0 win away to Alloa Athletic.
Falkirk ran out 5-1 winners against Albion Rovers at the Falkirk Stadium and Ayr United enjoyed a 3-0 triumph over Edinburgh City at Somerset Park.
Queen of the South beat Annan Athletic 3-1 at Galabank while Queen's Park play Motherwell at Firhill tomorrow, live on Premier Sports.
Clyde defeated Stranraer 1-0 at Broadwood and Morton beat East Kilbride 5-4 on penalties after a 0-0 draw at Cappielow.
St Mirren got the better of Dunfermline Athletic, winning 1-0 in Paisley, while Stenhousemuir were 2-1 winners away to Dumbarton.