Moffat netted from close range and added to that with a shot into the bottom left corner.
Aidan Nesbitt's strike halved Morton's deficit before the interval.
However, the visitors could not find an equaliser in the second period meaning the Pars went above Ayr United and Dumbarton.
Match ends, Dunfermline Athletic 2, Morton 1.
Second Half ends, Dunfermline Athletic 2, Morton 1.
Michael Moffat (Dunfermline Athletic) is shown the yellow card for a bad foul.
Jamie McDonagh (Morton) wins a free kick on the left wing.
Foul by Michael Moffat (Dunfermline Athletic).
Lee Ashcroft (Dunfermline Athletic) wins a free kick in the defensive half.
Foul by Thomas O'Ware (Morton).
Attempt missed. Ross Forbes (Morton) right footed shot from outside the box is high and wide to the right.
Foul by Lee Ashcroft (Dunfermline Athletic).
Kudus Oyenuga (Morton) wins a free kick in the defensive half.
Substitution, Dunfermline Athletic. Farid El Alagui replaces Nicky Clark.
Foul by Nathaniel Wedderburn (Dunfermline Athletic).
Caolan McAleer (Morton) wins a free kick in the attacking half.
Foul by Kallum Higginbotham (Dunfermline Athletic).
Mark Russell (Morton) wins a free kick on the right wing.
Foul by Paul McMullan (Dunfermline Athletic).
Jamie McDonagh (Morton) wins a free kick on the right wing.
Attempt saved. Nicky Clark (Dunfermline Athletic) right footed shot from the centre of the box is saved in the bottom right corner.
Gary Oliver (Morton) wins a free kick in the defensive half.
Foul by Rhys McCabe (Dunfermline Athletic).
Substitution, Morton. Caolan McAleer replaces Andy Murdoch.
Attempt missed. Kallum Higginbotham (Dunfermline Athletic) right footed shot from a difficult angle and long range on the left is high and wide to the right from a direct free kick.
Jamie McDonagh (Morton) is shown the yellow card for a bad foul.
Nicky Clark (Dunfermline Athletic) wins a free kick on the left wing.
Foul by Jamie McDonagh (Morton).
Foul by Ryan Williamson (Dunfermline Athletic).
Aidan Nesbitt (Morton) wins a free kick in the defensive half.
Nathaniel Wedderburn (Dunfermline Athletic) is shown the yellow card for a bad foul.
Foul by Nathaniel Wedderburn (Dunfermline Athletic).
Michael Tidser (Morton) wins a free kick in the defensive half.
Foul by Michael Moffat (Dunfermline Athletic).
Andy Murdoch (Morton) wins a free kick in the defensive half.
Corner,  Dunfermline Athletic. Conceded by Jamie McDonagh.
Lee Ashcroft (Dunfermline Athletic) wins a free kick on the right wing.
Foul by Kudus Oyenuga (Morton).
Foul by Rhys McCabe (Dunfermline Athletic).
Ross Forbes (Morton) wins a free kick in the defensive half.
Corner,  Dunfermline Athletic. Conceded by Andy Murdoch.
Foul by Nathaniel Wedderburn (Dunfermline Athletic).
Jamie McDonagh (Morton) wins a free kick on the right wing.
Michael Moffat's double moved Dunfermline Athletic off second bottom in the Scottish Championship as Greenock Morton lost in Fife.