Dundee United were on course for the win they needed to bounce back from their thrashing by Ayr United until a late goalkeeping error cost them two points in Greenock.
Matej Rakovan tried to punch a Michael Tidser free-kick away on 78 minutes but the ball skimmed off his gloves and presented Gregor Buchanan with a free header into an empty net.
Rakovan was one of four changes made to the starting line-up of eight days ago and another, Frederic Frans, had given United a first half lead with a looping header just after the half-hour mark.
Before their equaliser, Morton had a goal disallowed when Reghan Tumilty tapped in a Charlie Telfer shot that looked like it was heading inside the post.
Substitute Billy King wasted a great chance on a three-on-two counter-attack when he shot straight at keeper Derek Gaston but the hosts deserved their point.
And United, whose manager Robbie Neilson was sent to the stand late on, will have to wait at least another week for a return to winning ways.