Ex-Dundee striker Paul Ritchie headed just over after five minutes.
John Hillcoat dived low to deny Tom McManus the opener on 10 minutes before easily saving Iain Anderson's header.
Simon Lynch should have done better when he met Steven Robb's cross six yards from goal, but Dundee's top scorer could only guide the ball wide of the target 10 minutes before the break.
Hillcoat almost gifted Dundee a goal on the stroke of half-time when he allowed Paul Dixon's 35 yard drive to squirm from his grasp, but luckily for him the ball trickled the wrong side of the post.
Kelvin Jack saved the home side on the hour mark when he dived low to save Paul Ritchie's close-range shot, and moments later the visitors went close again when David White headed narrowly past.
With 20 minutes to go Kevin Byers crashed a right-foot effort off the post with Jack stranded.
Dundee went in front after 74 minutes when they were awarded a penalty out of nothing. Referee Crawford Allen awarded the kick for a definate foul by Hillcoat on Andy Ferguson. Lynch made no mistake form the spot, sending the keeper the wrong way.
Lynch should have made sure of the points five minutes later when he was clean through on goal, but from inside of the box he pulled his shot wide of the post.