The hosts eventually secured a pre-season showdown against St Mirren with a relatively comfortable 3-1 win at Cappielow last night - but not before the amateurs had given them an early scare.
After taking a seventh-minute lead through Alan Frizzell, who rifled high into the net after Michael Tidser's low cross found him unmarked eight yards out, Ton were given a shock to the system when Gourock fired back five minutes later.
Midfielder Neil McCole picked himself up after he was fouled inside the box by Paul Di Giacomo and calmly stroked a fine spot-kick past goalkeeper Matt McGinlay to quickly restore parity.
The plucky Premier Division 2 side almost went ahead two minutes later when Shaun Finlay headed Darren McGregor's drilled cross narrowly wide of the upright.
But Ton quickly recovered their composure and Lewis Hawke met a fantastic pin-point Tidser centre with a textbook downwards header to restore their lead in the 25th minute.
Unlike most of Morton's semi-final opponents in recent years, Gourock refused to buckle and performed admirably, with Sean Marner and Paul Flynn forcing stops from McGinlay and his replacement Shahrukh Khatri.
Meanwhile, at the other end, keeper Michael Ptolomey kept Ton at bay with a string of impressive saves to deny Peter Weatherson, Frizzell, Di Giacomo, Joel Kasubandi and Dylan McLaughlin.
However, substitute Weatherson finally dashed any hopes of an upset in the 71st minute when he curled an unstoppable 25-yard free kick over the wall and into Ptolomey's top right-hand corner.
Morton manager Allan Moore said: "We didn't take some of the opportunities that we should have.
"I didn't go into the dressing room after the game, but Mark McNally would've told them that it doesn't matter what game it is, they need to put those chances in the back of the net.
"But these games are a potential banana skin, so we were just glad to get through without picking up any more injuries.
"It was a good run out for our young boys. We sprinkled the team with some experienced players to play beside them and give them a view of what we're looking for.
"Overall, it was a positive result and we can now look forward to the game against St Mirren at the beginning of next season."
Subs not used: MacDonald.
Gourock Thistle (3-5-2): Ptolomey; MacGuire (Hastings, 68), McColl, Gunn; Flynn, McGregor (Campbell, 71), N McCole (Patrick, 58), D McCole, Finlay (Carby, 35); Miller, Marner (McKellar, 66).
Referee: Mark McLean.