Stewart Kean headed Peter Weatherson's cross into the net to level for Morton in the 59th minute.
But Marc McKenzie set up John Dempster to fire home from 12 yards 18 minutes from time to ensure Cowden took the three points back to Central Park.
Morton: Stewart, Evans, Smyth, McCaffrey, Bachirou (Jenkins 56), Tidser, O'Brien, Holmes, Fitzharris (Graham 81), Weatherson (Kelbie 77), Kean. Subs Not Used: Cuthbert, Greacen.
Goals: Kean 59.
Cowdenbeath: Hay, Baxter, Adamson, Armstrong (Mbu 46), Linton, Ramsay, Winter, O'Brien (Sibanda 75), McKenzie, Cameron, Dempster (Stewart 78). Subs Not Used: Roy, Byren.
Goals: Ramsay 52, Dempster 72.
Att: 1,839
Ref: B Colvin
Source: BBC Sport
Source: BBC Sport