Dundee dropped to sixth place in the Championship after a fifth successive game without a win.
The Dark Blues were at least able to end their three-game goal drought when a long ball from Christophe Berra, making his debut after his move from Hearts, should have been cleared by Stephen McGinty, but fell perfectly for Kane Hemmings, who drilled the ball home from eight yards after 12 minutes.
Morton, who have only managed four victories in their last 15 clashes with the Dundee, levelled after 26 minutes when Shaun Byrne conceded a needless free-kick at the edge of the area. Nicky Cadden’s 20-yard shot took a wicked deflection, but was well saved by goalkeeper Conor Hazard, but he failed to hold and Jim McAlister followed up to clip home the rebound.
Two minutes later Ton almost got a second, but Kalvin Orsi’s curling 25-yard effort dipped over.
The hosts, with just three wins in eight games, grew with confidence and one minute from the break Luca Colville was given too much space to stride forward, but his 20-yard driven effort was beaten away by Hazard.
Seconds later Berra appeared to pull down Orsi, but Bobby Madden rejected Morton’s penalty claims.
Morton were reduced to ten men on 58 minutes after Kyle Jacobs was shown a second yellow card for a crunching tackle on Hemmings.