The Dark Blues went close on 12 minutes when Steven Robb's cross deflected off the foot of Murray Henderson before rebounding off the crossbar to Garry Brady, who crashed a shot into the side-netting.
Midway through the half Barry-John Corr did well to save Simon Lynch's point-blank effort and moments later easily saved Brady's 15-yard shot.
Michael Moore fired high and wide of the target for the visitors before Kevin McDonald pulled his effort well wide for the home team a few minutes before the interval.
On the stroke of half-time Craig O'Reilly's 25-yard effort went just past the post.
Dundee took the lead in the second minute of the second half when young striker O'Reilly fired home from eight yards after a good run and cross by Paul Dixon.
Henderson then attempted to level for the visitors from 35 yards, but his shot went high into the empty stand behind the goal.
With 70 minutes on the clock, Dundee's 17-year-old keeper Scott Murray dived full length to turn Lee Sharp's net bound free-kick around the post.
Lynch made it 2-0 to the hosts with 10 minutes remaining. It was a great individual effort which saw him cut in from the left and go past two defenders before firing in a left-foot shot from 18 yards which flew past John-Corr with the aid of a slight deflection.