Cove Rangers were undone by a late Robbie Crawford goal at the Balmoral Stadium as Morton emerged with all three points.
Morton had led from the closing stages of the first half after Jaze Kabia’s goal but substitute Leighton McIntosh looked to have rescued a point for the home side.
But Crawford stepped up to be the hero in the closing stages as Dougie Imrie’s side extended their unbeaten run to eight games.
Cove made one change from the 1-1 draw with Arbroath, with Luis Longstaff returning to the lineup for the suspended Mitch Megginson.
The home side had the wind at their backs in the first half and tried to make it count early, with Fraser Fyvie fizzing in a low free-kick to a dangerous area which Brian Schwake gathered.
Ton stopper Schwake kept his side level with a superb stop on 21 minutes, after Mark Reynolds had recycled the ball into the penalty area to pick out fellow centre-back Sanders. His header was firm and low but Schwake stuck out his right hand to keep it out.
Morton had struggled with the conditions for much of the first period but should have hit the target 10 minutes before the break.
Kabia won back possession on the right and crossed, with Stuart McKenzie palming the ball clear. It broke to left-back Lewis Strapp, who cleared the bar and the stand with his effort.
But the visitors were more accurate with their next attempt as Jai Quitongo profited from some slack defending to find Kabia, who crashed the ball beyond McKenzie.
Cove could have been level on the stroke of half-time after Fyvie glanced Luis Longstaff’s free-kick inches wide of the far post.
Morton almost doubled their lead in spectacular fashion on 56 minutes, with Robbie Muirhead going for goal from a free-kick 30 yards out. McKenzie watched on as the ball cannoned back off the frame of the goal.
The Cove goalkeeper was required 10 minutes later after Robbie Crawford worked space for a shot from distance and McKenzie sprung to his right to parry.
The game had descended into a scrappy affair and was not helped by the winds, which played a part in Cove’s leveller with seven minutes to go.
Shay Logan collected Max Johnston’s over-hit cross and returned the ball towards the back post, with Schwake misjudging as he appeared to wait for it to go out of play. It instead flicked off the inside of the post back into play and McIntosh nodded in.
But the hosts were unable to hold on to their lead as Crwaford smashed through a crowded penalty area with three minutes to go to seal the points.
COVE RANGERS (4-4-1-1) – McKenzie 6; Johnston 6, Sanders 5, Reynolds 5, Logan 5, Longstaff 5 (Masson 59), Yule 6, Gilmour 6, Scully 6 (Leitch 80), Fyvie 6, McDonagh 5 (McIntosh 65). Subs not used – Gourlay, Ross, Neill, Vigurs, Dunne, Leitch.
MORTON (4-2-3-1) – Schwake 6; Grimshaw 6, Baird 6, Ambrose 6, Strapp 7, Blues 6, Crawford 6, Quitongo 6, McGrattan 6 (King 77), Kabia 7 (Pignatiello 77), Muirhead 6. Subs not used – O’Connor, Hynes, McGregor, Lithgow.
Referee – Peter Stuart 6.
Attendance – 870.
Player of the match – Lewis Strapp.