Paul Hartley's side had the opportunity to take all three points through Gary Harkins' 59th minute penalty, but substitute Thistle goalkeeper Paul Gallacher saved his effort with his first piece of action after replacing the red-carded Scott Fox.
Thistle had surged into an early lead thanks to Gary Fraser's 25-yard drive, but were hauled back when substitute Craig Wighton stepped off the bench to level matters for the Tayside club with a well-taken goal within 10 minutes of the restart.
Dundee pushed for a winner towards the end but could not make the extra man count against Alan Archibald's men.
Thistle, who arrived on Tayside buoyed by the midweek 4-0 win over Ross County, were ahead in just four minutes as Fraser picked up a pass from Steven Lawless and wasted no time firing an unstoppable shot high into the top corner past the helpless Kyle Letheren.
It was some strike from Fraser, whose confidence was high going into the game having netted in the County game a few days earlier.
That gave the visitors a massive boost and they threatened to increase their lead just a few minutes later when Lawless crossed for Ryan Stevenson, but his shot was blocked by Dundee defender James McPake.
Dundee were struggling to get going although they did receive a bit of heart when Peter MacDonald's free-kick from 22-yards went just past Jags keeper Fox's right-hand post.
Then, in the 16th minute, Partick were forced into a change when former Rangers youngster Jordan McMillan hobbled off injured and was replaced by Dale Keenan.
However, the hosts remained on top and midway through the first half they came close to going further in front.
Stephen O'Donnell got down the right and his cross found its way to Stuart Bannigan who got in front of Dundee defender Paul McGinn, but Letheren in the home goal was equal to the task.
At the other end, Paul McGowan broke forward for the hosts but there were no takers for his cross into the box, which was the last real action of the first half.
The second half started with chances at both ends as Gary Harkins had a free-kick for the home team which hit the bar.
Shortly after, Kallum Higginbotham unleashed a powerful shot from distance which Letheren was pleased to see go past his far post.
However, Dundee were level in the 54th minute. Jim McAlister slid a pass through to substitute Wighton, who kept his cool to send his finish though Fox's legs to make it 1-1.
Dundee started to put the pressure on and in the 58th minute Thistle defender Conrad Balatoni was booked for pulling back Peter MacDonald.
Soon after the hosts were awarded a penalty when Daniel Seaborne was adjudged to have fouled James McPake as the pair went for a high ball inside the box - and Fox was sent off after what appeared to be a clash with McGowan as tempers flared.
Harkins stepped up to take the kick - but substitute goal-keeper Paul Gallacher defied him from the spot, saving the effort with his left leg.
Dundee kept going in search of a winner and in the 70th minute Harkins' cross was met by MacDonald, whose back-post header was well saved by Gallacher.
Thistle were pegged back for long spells but dug deep with a man less to come away with a point.