A 20-yard strike from Richard Foster was deflected just past the post for a corner with six minutes played.
Five minutes later a cracking 22-yard effort from Russell Anderson was well saved by Derek Soutar.
The home team had the ball in the back of the net on 32 minutes but Neil Barrett's strike was ruled out for offside.
Ten minutes before half-time John Sutton's low drive was turned around the post by Ryan Esson in the visitors goal.
Barrett's dipping shot from 35 yards went just past the post with three minutes to go before half-time.
Darren Mackie had the first effort at goal in the second half but his shot from the edge of the box was saved by Soutar.
Dundee opened the scoring with 57 showing on the clock and it was Barrett who collected a pass from Stephen McNally inside the danger area and from eight yards the midfielder drove the ball low past Esson.
Aberdeen were reduced to ten men on 67 minutes when substitute Steven Craig was ordered off from aiming a kick at Barrett.
Fifteen minutes form time Garry Brady had a great chance to double Dundee's lead but he shot straight at the keeper from eight yards.
Aberdeen almost drew level with six minutes remaining when first Chris Clark was denied by Soutar and when the ball broke it fell to Anderson who crashed a low drive against the post.
In a frantic final few minutes Aberdeen threw everything at the home team and despite creating a couple of very good chances, could not grab the equaliser.