The Tangerines appeared on their way to defeat when faced with a 3-1 deficit but two goals during the closing stages earned them a share of the spoils.
It took the visitors just six minutes to take the lead at Tannadice, with Frederic Frans rising above the Dundee defence to powerfully head home a corner delivered by James Penrice, who was making his first-team debut at the age of 17.
Dundee responded positively to going behind but both Billy McKay and Simon Murray were unable to find the target while midway through the first half, McKay forced a solid save out of goalkeeper Ryan Scully.
Dundee again continued to threaten after the restart with McKay sending a header wide of the target, but with 23 minutes remaining they finally got on the scoresheet when Liam Lindsay put through his own net
However, their encouragement was short-lived as Partick regained their two-goal cushion through Ryan Edwards, who was in the right place at the right time to benefit from some terrific work from Amoo to head home at the back post.
Dundee reduced the deficit for a second time when Edward Ofere was afforded too much time and space inside the penalty area before firing home and it set up a grandstand finish with Partick looking to hang on to a result that would see the club move into eighth place.
The home side had the last laugh, however, as Justin Johnson found the corner to earn the beleaguered club their fourth point in the space of four days.