A goal from Scott McManus and John Sweeney's penalty were sufficient to see off the League One side.
Ferguson said: "It was a good start for us, but we were very poor in the second half and the players have had a rollicking for that. We are really still only midway through our pre-season so there is still room for improvement."
Brora Rangers produced the shock of the round by knocking out Stenhousemuir.The Highland League champions led through Zander Sutherland's first-half opener, but Martin Grehan's equaliser took the tie into extra-time.
However, Brora dominated the overtime during which Sutherland completed a memorable hat-trick.
Another shock came at Central Park where Championship side Cowdenbeath fell to League One Brechin City 3-1 after extra-time.
Dumbarton surrendered a two-goal cushion at Stair Park before Sean Winter's injury-time winner for League One Stranraer.
Similarly, it was the side from a lower division who came out on top in the Lanarkshire derby, Albion Rovers securing a 4-2 penalty shootout win at Airdrie following a 2-2 draw after extra-time.
Lowland League champions Spartans gave Morton a run for their money before going down to Cameron O'Neil's winner at Cappielow.
East Stirling restricted derby rivals Falkirk to a one-goal lead at the break before Rory Loy and Blair Alston both grabbed doubles as the Bairns turned on the style to win 7-1 at Ochilview.
Championship class also told for Alloa Athletic who had a comfortable 4-1 win at Arbroath while East Fife (2-1 v Forfar), Stirling Albion (3-0 at Elgin) and Peterhead (3-0 at Montrose) also made it through.
The only all-Championship tie produced real fireworks as Queen of the South and Livingston traded goals liberally in a thriller at Palmerston Park. Lions substitute David Robertson eventually got the decisive seventh goal five minutes from the end of extra-time.