Stranraer were the first to threaten with David Graham sending a low shot wide.
Morton picked up immediately but Jim McAlistair's right-footed cross was poor and went behind.
In the eighth minute the 18-year-old put his left-foot too much better effect.
Morton was still buzzing around some 20 yards from Stranraer's goal when the ball bounced fortuitously into McAlistair's path and the youngster advances a couple of paces before dispatching a spectacular shot into the top corner of Kenny Meechan's net.
Steven Swift could have equalized from a Lee Sharp corner, but glanced his header wide.
The action quickly transferred to the other end and Alex Williams volleyed over from Warren Hawke's clever chested pass.
Stranraer were millimetres from levelling when Fraser Wright whipped in a fine cross and David Turnbull's header struck the foot of the post and fell kindly into the arms of Craig Coyle.
Four minutes from the interval, Stewart Greacen fired in a cross from the right and Stuart McCluskey cracked it across goal where Wright deflected it past Meechan.
Seven minutes after the break, Sharp flung in a free-kick to the far post and Murray Henderson slid in to knock the ball pass Craig Coyle from seven yards.
A minute later Derek Wingate fouled Hawke in the box and Williams restored the two-goal cushion.
Stranraer battled on and were unlucky when Graham blasted the ball off Coyle after Sharp had hit a post and again when Derek Collins managed to get his head in the way of another Graham drive to deflect the ball for a corner.