Terrors manager Jackie McNamara was returning to the club that he left United last season, and it was his side that had the clearest of the chances in the opening 30 minutes.
Scottish international Goodwillie threatened again prior to the break, but could not connect cleanly with his shot from a promising position on the edge of the box.
The hosts forged some chances of their own as the first period went on and would have been ahead had it not been for a superb stop from Radoslaw Cierzniak, who clawed a deflected cross away from his goalline.
Gary Mackay-Steven came off the visitors' bench for the second half and was lively down the United flanks, but could not help his side muster an opening goal.
In fact, there were few more opportunities until the dying stages of the contest when Partick forward John Baird's header drifted agonisingly past the post, before Cierzniak denied Ross Forbes with the last kick of the game.