The Jugs struck the fatal blow in first half stoppage-time when Doolan swivelled inside the penalty area and shot across Nico Caraux and inside the post
Conrad Balatoni passed up a fine chance to put the visitors in front with a close-range header on 17 minutes when he failed to make a proper connection and looped his effort over the top from seven yards.
Just three minutes from the interval Morton scorned an ideal chance to break the deadlock after Thistle's Abdul Osman clattered Lee Kilday on the right to concede a free-kick.
Reece Hands sent in a terrific delivery to pick out unmarked Stefan McCluskey at the back post, but he nodded beyond the far post from six yards.
Thistle should have had the tie wrapped up after 53 minutes when Steve Lawless robbed Lee Kilday and fed Stephen O'Donnell who in turn then found Kallum Higginbotham on the left of the box, but his ferocious shot was brilliantly beaten out by keeper Caraux.
And a minute later, Stefan McCluskey dispossessed last-man Balatoni at the edge of the box and curled his shot past the advancing Paul Gallacher and wide of the post.