David Clarkson, Gary Harkins and Greg Stewart put Dundee in a comfortable position inside an hour before Henrik Ojamaa pulled a goal back for the home team.
Clarkson found the top corner of the net after just three minutes when the former Motherwell midfielder hit a superb shot on the turn.
Harkins doubled Dundee’s advantage after half-an-hour with another firm drive from inside the penalty area, leaving Dan Twardzik with no chance.
Dundee secured the points on 59 minutes when Motherwell were carved open with ease yet again and the unmarked Stewart had the simple task of tapping home after Clarkson set up the opportunity.
Motherwell pulled a goal back three minutes later when a Lionel Ainsworth corner was flicked on at the near post and Ojamaa stabbed the ball over the line after Kyle Letheren pushed the initial effort against the underside of the bar.
The home side had a chance to respond to the early concession when Ojamaa was put clear in the box but his shot was saved by Letheren.
Simon Ramsden and Stephen McManus then both had efforts at goal inside a minute, which were saved by Letheren.
But Paul McGowan came close to adding a third in first-half stoppage time when his sweeping drive was saved by the diving Twardzik.