Seven minutes after the interval Weatherson collected Allan Jenkins pass and fired into the bottom corner.
A tense finish ensued when Kevin Smith netted a penalty spot after Gregory Tade was fouled by Dominic Shimmin.
Raith Rovers: McGurn, Wilson, Ellis, Campbell, Murray, Walker, Russell (Darren Smith 68), Simmons, Kevin Smith, Tade (Weir 85), Ferry (Williamson 68). Subs Not Used: O'Connor, Sloan.
Booked: Williamson. Goal: Kevin Smith 79 pen.
Morton: Stewart, McGuffie, Reid, Shimmin, Greacen, Tidser (Paartalu 55), Finlayson, Jenkins, Weatherson, Graham (Russell 79), McAlister. Subs Not Used: Cuthbert, MacGregor, Monti.
Booked: Tidser, McGuffie. Goal: Weatherson 33, 52.
Att: 1,702
Ref: C Allan
Source: BBC Sport
Source: BBC Sport