Morton took the lead after only seven minutes. Captain Chris Miller and Michael Tidser combined to put Jack Iredale scurrying down the left. His low driven cross was met at the back post by Bob McHugh, who turned the ball past Brett Long.
But East Fife were level within three minutes when Liam Watt cut the ball back from the right for Anton Dowds, unmarked in front of goal, to lash the ball past goalkeeper Derek Gaston.
Morton almost regained their lead in 19 minutes when McHugh drove forward from midfield, but his 18-yard effort was superbly held by Long.
The Greenock men dominated the second period but it was the Bayview side who grabbed the winner in 76 minutes when a corner from the right found Watson lurking at the back post to drill the ball low past Gaston.
Morton thought they had levelled in 84 minutes when Charlie Telfer’s free kick from the right evaded everyone except Watson, who knocked the ball clear.