A late first-half strike from Kris Doolan earned Partick a 1-0 win in the Scottish League Cup but they were made to fight all the way by Morton on Tuesday night.
Doolan scored in the 45th minute to send Partick into the draw for the third round against a League One outfit who only named five substitutes but gave their Premiership opponents a real run for their money.
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.
Then Gary Fraser blazed high and wide from distance as Thistle struggled to break down Morton.
And there were no real chances until the 25th minute when Morton goalkeeper Nico Caraux made the first serious save of the match to smother a 20-yard drive at his far post from Steve Lawless.
Just three minutes from the interval and Ton 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 teased in a terrific ball to pick out unmarked Stefan McCluskey at the back post, but from six yards his attempted header skidded beyond the far post.
And Jags punished them with the opener in first half added-time when Doolan gathered a harmless looking ball in the box before suddenly swivelling to find space and a shot at goal which he placed across Caraux and inside the left post from 15 yards.
Thistle should have had the tie wrapped up after 53 minutes when Lawless robbed Lee Kilday of possession and fed Stephen O'Donnell who 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 blew a great chance to level the clash after dispossessing last-man Balatoni at the edge of the box.
But after closing in on goal, and luring keeper Paul Gallacher from his line, McCluskey curled a 15-yard shot around the far post.
Doolan came close to a second goal for Partick with a sweet, swerving shot which slipped inches beyond the far post before McCluskey latched onto a Declan McManus header at the other end only to be spent sprawling by keeper Gallacher.
Ton fans were on their feet screaming for a penalty while McCluskey squared up to the Jags keeper before being booked for diving after 71 minutes by referee Stephen Finnie.