The Greenock side took the lead on 23 minutes when Aidan Ferris applied the finishing touch to a Dylan McLaughlin corner.
Port's Graham Black levelled things up just three minutes later, sending David Hutton the wrong way with a confidently-struck spot-kick.
But the visitors restored their lead one minute before the break when McLaughlin twisted and turned on the edge of the box before curling a sublime effort into the net via the underside of the crossbar.
Under-17s striker Jordan Cairnie, one of 10 half-time changes for Ton, showed both an impressive turn of pace and a cool head when he raced clear to stroke home the third on 52 minutes.
Motherwell youth coach Jonatan Johansson was in attendance to cast an eye over Fir Park trialist Aidan Fulton.
And he couldn't have failed to be impressed when the wideman whipped home a fantastic fourth.
The Undertakers wrapped up the scoring in the last minute, Ton loanee Martin Maguire forcing home a late corner to cap his encouraging display in the Port defence.
Morton (4-4-2): Hutton (McFarlane); Boyd (Lamont), Irvine (Tennent), Jamieson (Verlaque), Ramsay
(Knight); D. McLaughlin (Russell), Keogh (McLean), McGowan [Trialist] (Lynch), McDaid (Fulton); Frizzell (Cairnie), Ferris (Hawke).