Cowdenbeath 0-2 Dundee
Craig Beattie's wonderful goal helped Dundee record back-to-back victories for the first time this season.
The former Celtic striker expertly lobbed stranded Cowdenbeath goalkeeper Thomas Flynn from 35 yards out to give the visitors the lead.
Dundee consolidated in the second half with Declan Gallacher and Peter MacDonald both being denied by Flynn.
And the game was wrapped up when Conroy slotted home from inside the area following good work from MacDonald.