Elsewhere Inverness Caley Thistle will face the other Highland League side left in the competition, Brechin City, at Caledonian Stadium.
League One Peterhead have been paired with north rivals Elgin City of League Two with the sides meeting at Balmoor.
Cove Rangers will tackle fellow Championship outfit Raith Rovers at the Balmoral Stadium.
Ties will be played on the weekend of September 24 and 25.
The third round draw in full is as follows: Buckie Thistle v Linfield, Cliftonville v Queen’s Park, New Saints v Dundee, Clyde v Caernarfon Town, Peterhead v Elgin, Falkirk v Partick Thistle, Cove Rangers v Raith Rovers, Dunfermline Athletic v Celtic B, Arbroath v East Fife, Morton v Ayr United, Montrose v Kilmarnock B, Alloa Athletic v Airdrieonians, Queen of the South v FC Edinburgh, Annan Athletic v Kelty Hearts, Rangers B v Hamilton Academical.
