It took the visitors 15 minutes to turn their clear superiority into an attempt on goal, but what an attempt it was as James Stevenson forced keeper David Orr to dive full length to push his long-range free-kick over the bar.
Cowdenbeath were being outclassed although striker Liam Buchanan did test goalkeeper Stefan Gonet with an improvised header.
Otherwise it was one-way traffic, Paul McGowan wasting a free header and Chris Millar also of target with a half volley.
Morton's constant pressure finally paid off after 33 minutes when defender Stewart Greacen nodded home Ryan Harding's cross at the back post.
The Greenock club went further ahead shortly before half-time as James McAlistair fired a low shot into the far corner.
Further goals looked certain as Harding headed wide and McGowan hit the post but Cowdenbeath then pulled one back after 55 minutes.
Martyn Fotheringham split the away defence and Buchanan crossed for Markus Paatelainen to bundle the ball over the line at the second attempt.
An unlikely comeback briefly seemed possible and Pat Clarke's deflected effort caused an anxious moment for Gonet, but the visitors regained some control and survived the closing minutes.