The visitors went behind in the second minute. Jim McAlister broke free on the left and his chip was palmed aside by the uncertain Scott Black. The ball broke off a defender to Jamie Stevenson who volleyed it high into the net from six yards.
Worse followed five minutes later when Peter Weatherson headed in a Stevenson free-kick with considerable ease.
"Black tipped a low Paul McGowan effort past a post before Stranraer had their first attempt. Paul McMullen, who had shown menace with two earlier crosses, slung over a fine one for Michael Moore, but the combative striker's header flew over the bar.
Moore had an angled drive saved, but Black almost gifted Morton a third when he fumbled a Chris Millar shot. Marc McAusland then hit Black with his attempted clearance from McAlister's follow-up.
Stranraer were competing robustly and referee Allan booked five of their players before the interval. William Snowdon headed a Mark Crilly cross wide as the visitors gradually eased the pressure.
The second half saw Stranraer make a valiant effort to reduce the arrears and Moore headed over a McMullan free-kick, had a shot saved by David McGurn and saw a low drive whistle inches past.
The league leaders clinched the points when Chris Millar's cross was headed past Black by McGowan.
David McGregor sent in a low, skimming cross which just eluded Millar and crept past the grasping Black's left-hand post.