A dramatic late comeback fell just short as we battled back from 4-1 down with just eight minutes to go.
There was one change in the starting line up from the side which had beat East Stirlingshire with Thomas O’Brien replacing Lewis Milne.
It took the lively looking visitors just 12 minutes to break the deadlock when Peter Weatherson got on the end of a Taggart cross and sent his header past Flynn from six yards.
Morton had a few more sieges on the home goal before Weatherson got his second goal of the afternoon on 25 minutes when the Cowden defence failed to clear a Graham cross and Campbell jinxed into the box before Weatherson fired the ball home.
The visitors almost went 3-0 up 12 minutes from the break when Taggart crashed a powerful shot from 35 yards which crashed off the underside of the bar before bouncing into the hands of Flynn.
Five minutes from the interval we pulled a goal back. Linton took our first corner of the afternoon and the ball was cleared as far as Stevenson on the edge of the box who lobbed the ball to John Armstrong who headed home from six yards.
However within sixty seconds of the restart Morton had restored their two goal advantage.
A cross from Graham was knocked back by O’Brien into the path of the former Central Park who hit the ball high into the net via the underside of the bar.
After being provider David Graham got the visitors fourth goal of the afternoon on 54 minutes when he ran into the Cowden box before firing past Flynn.
With a routine away win looking on the cards a Cowden revival brightened up the closing ten minutes.
Miller broke though on goal but his shot was tipped over the goal by Gaston however two minutes later we got a second goal when substitute Mark Ramsay slotted a Linton cross home at the back post.
Four minutes from time Miller was once again denied a goal when Reid cleared his net bound shot off the line.
However with almost the last touch of the game Jamie Stevenson got a third goal in injury time when he nodded home from a long Linton throw in.