Darren Barr had an effort saved as the Sons tried to force a lead.
Morton's Ross Forbes hit the post with 10 minutes left, Denny Johnstone firing the rebound wide.
Gregor Buchanan went close moments later at the other end but Derek Gaston was alert to smother the chance.
Dumbarton remain just two points clear of Livingston in the relegation play-off place after drawing at home to Greenock Morton in the Scottish Championship.