Partick took the lead with a Gary Fraser strike and Dundee levelled in the early stages of the second half through Craig Wighton, before substitute keeper Paul Gallacher saved a Gary Harkins penalty seconds after Thistle keeper Scott Fox had been ordered off.
Fraser gave the visitors the ideal start when he stepped on to a cutback from Steven Lawless and crashed his shot into the top left corner past Kyle Letheren after just four minutes.
Dundee substitute Wighton (54) kept his composure to roll the ball under the advancing Fox after Jim McAlister set up the chance with a great solo run through the Thistle defence.
And Dundee had an opportunity to take the lead six minutes later when James McPake was brought down in the box by Danny Seaborne and referee Craig Charleston awarded a spot-kick.
Charleston also produced a red card moments after that when Thistle keeper Fox clashed with Dundee midfielder Paul McGowan.
When order was restored Harkins stepped up to take the penalty, but Gallacher’s first action was to make a save with his left leg and keep Thistle on level terms.
After their early goal Partick had threatened again when another shot from Fraser was held by Letheren shortly before the Dundee keeper denied Stuart Brannigan with his feet.
And the second half started with chances at both ends as Harkins had a free-kick for Dundee which hit the bar and shortly after, Kallum Higginbotham unleashed a powerful shot from distance which Letheren was pleased to see go past his far post.
Dundee kept going in search of a winner against the 10 men and a cross from Harkins' cross was met by Peter MacDonald, whose back-post header was well saved by Gallacher.
Although Thistle were pegged back for long spells they dug deep to earn a deserved point against a Dundee team seeking their first win after starting the season with three draws.