Dundee came from two goals down to keep alive their slim hopes of winning the First Division title.
David Witteveen gave Morton a deserved lead on 41 minutes from close range as Dundee struggled to create any chances.
Carlo Monti doubled Morton's lead from the penalty spot on 64 minutes, before Gary Harkins pulled one back three minutes with a good strike.
Five minutes before the end, Leigh Griffiths rescued a point for Dundee with a fantastic curled free-kick.
The result leaves Dundee still able, arithmetically, to win promotion to the Scottish Premier League.
They lie nine points behind leaders Inverness Caledonian Thistle with nine points to play for, although the Highlanders have a far superior goal difference.
After 45 minutes at Cappielow, however, it looked like the Dark Blues' championship aspirations would end on Saturday.
Gordon Chisholm's men looked flat and failed to trouble home goalkeeper Colin Stewart even once.
Instead, it was Morton - still desperate to get clear of the relegation play-offs - who had more energy, battled harder and looked more likely to score.
Michael Tidser, Erik Paartalu and Ryan McGuffie all made Tony Bullock produce saves before Witteveen reacted quickest to a header that came off the bar to open the scoring.
Morton's Peter Weatherson broke free moments after the restart for the chance to make it 2-0 but lacked composure when one-on-one with Bullock and pulled his effort badly wide.
But Morton did make it two from the spot after Witteveen outpaced Lauchlan, was tripped and Monti drove the ball down the centre of the goal from the spot.
A number of the travelling support headed for the exits at that point, but Harkins got Dundee back into the game by skipping past his man in the box and rifling a shot across Stewart inside the post.
And, just five minutes from securing a vital win in their quest to safeguard their First Division status, Morton conceded a free-kick 24 yards from goal.
Griffiths hobbled back on the pitch from receiving treatment to an injury to strike a sweet left-footed shot over the wall into the net.
Morton manager James Grady:
"When you're 2-0 up, I think you should be able to see out the game
"I'm gutted for the guys because we played really good stuff and I think we deserved the three points.
"But I think some the decisions went against us today.
"We've got good players here. They just need to believe in themselves more.
"We out-passed Dundee for about an hour. We had chances and didn't take them and that's come back to haunt us."
Dundee manager Gordon Chisholm:
"Considering what we were playing for, I was really disappointed.
"The players were told the one thing they would get is a physical game and they didn't handle that.
"I thought we got bullied. Unbelievably, we weren't up for the fight in the first 45.
"It wasn't until we went 2-0 down that we thought we better start playing.
"But to show that lack of urgency and determination like that in the first half is unacceptable."
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