The first half was rather one-sided as the home side, with David Hopkin still in temporary charge, struggled to find their feet.
Craig Beattie, with the help of a deflection off Mark McLaughlin, put Dundee in front after 17 minutes following an excellent through ball from Peter MacDonald.
A couple of fine saves from Nicolas Caraux prevented Dundee from adding to their lead but the goalkeeper was left with no chance on the stroke of half-time when Declan Gallagher headed home Nicky Riley's corner.
Caraux made another good save early in the second half, this time denying Jim McAlister. Dougie Imrie pulled a goal back with a left-foot shot just before the hour and Morton had chances to salvage something from the game but Dundee held on to take all three points.
Hamilton Academical lost their position as outright leaders when they went down 1-0 to Alloa Athletic at New Douglas Park.
Darryl Meggat headed the only goal two minutes into the second half as Paul Hartley's team saw off a misfiring Hamilton side.
The hosts had the better of the first half but Scott Bain, in the Alloa goal, stood up to everything that came his way and the visitors made the breakthrough two minutes into the second half when Meggat, left unmarked at the back post, connected with Kevin Cawley's cross to score.
Andy Ryan stretched Bain with a couple of decent efforts after that - one which the goalkeeper turned on to a post - but Alloa held out.
Falkirk maintained their challenge with a comfortable 3-1 home win over a disappointing Raith Rovers. The result saw Gary Holt's side leapfrog their Kirkcaldy opponents and assume third place on goal difference.
Craig Sibbald headed the opener past Raith goalkeeper Ross Laidlaw from Stephen Kingsley's cross after only three minutes and Falkirk increased their advantage midway through the first half when Conor McGrandles shot home after Rory Loy's initial effort had been blocked.
Kingsley wrapped up the points with a close-range effort early in the second half before Greig Spence pulled back a consolation goal.
Jimmy Nicholl's first match on his return as Cowdenbeath manager was going to plan for 75 minutes as his team led Livingston 2-0 through a Kane Hemmings double, but calamity followed as the visitors to Central Park snatched a 3-2 win.
Hemmings broke the deadlock after 56 minutes with a shot from six yards. He added his second three minutes later after finding himself one-on-one with Livingston goalkeeper Darren Jamieson.
However, Marc McNulty headed in a Danny Denholm cross to start the fightback after 77 minutes and Andy Barrowman equalised with another header 10 minutes later before McNulty netted again in injury-time.
Queen of the South beat Dumbarton 1-0 at Bet Butler Stadium, the only goal coming early in the game when Iain Russell's shot went in off Andy Graham, leaving home goalkeeper Jamie Ewings stranded.