The rout began in the fifth minute when James Stevenson played a free-kick into the path of full-back Peter Weatherson and he drove a low shot through a ruck of players from 16 yards.
Morton, who had lost on all their previous visits, extended their lead after 18 minutes. The goal was made by the pace of James McAlistair, who burst down the left wing before squaring the ball to Paul McGowan, and the youngster, on loan from Celtic, swivelled to score from six yards out.
Peterhead could have scored in the 25th minute. Chris Hegarty played the ball forward quickly from a free- kick, but Bobby Linn fired his first-time effort high and wide of the target.
Any chance of a comeback was soon ended when Morton scored a third in the 34th minute. Poor defending from a corner found McGowan in space at the back post, and he scored with ease from two yards.
The fourth goal was a well-taken effort right on the stroke of half-time. Chris Millar took the ball in his stride 20 yards out, and the ball flew into the net by the right-hand post.
The second half was predictably a quieter affair. Peterhead went close with their only real shot on target in the 56th minute, Dougie Cameron shooting off the underside of the crossbar.