Hibs looked like dropping into serious trouble when Kris Doolan fired Thistle into an early lead and they went on to waste a series of chances to equalise, with visiting keeper Paul Gallacher in inspired form.
But Terry Butcher's side grabbed a point two minutes from time to remain in eighth position, with four teams still fighting to avoid finishing in 11th and having to go through the playoffs to avoid dropping into the Championship.
Hibs started brightly, with Michael Nelson heading wide in the first couple of minutes, but they fell behind on eight when Doolan got to a cross just before home keeper Ben Williams to open the scoring.
The home side went in search of a quick equaliser and came close when a shot from Jason Cummings went inches wide of goal.
Doolan was forced off injured before the 20th minute, and while Thistle were getting their shape back together Gallacher made a fine save to keep out Liam Craig.
Gallacher had a busy few minutes after that save, denying Nelson twice, with the ball coming back off the post as well during a goalmouth scramble in the wake of his first save.
Ryan McGivern headed wide for Hibs but they went in a goal down and the pattern continued after the break, with Cummings heading into the side-netting as the home side pressed.
Gallacher kept the ball out in another melee caused by a Nelson header and a series of corners were wasted by Hibs in their increasingly desperate search for an equaliser.
Hibs were reduced to 10 men inside the final 10 minutes when Nelson was shown a second yellow card, but they continued to play on the front foot and Gallacher again came to Partick's rescue by denying Craig.
However, their perseverance finally paid off two minutes from time when Stanton headed past Gallacher and secure a massive point for the Hibees.
The draw leaves Hibs and Partick level on 35 points, in eighth and ninth, with a couple of games remaining although Kilmarnock could move past both teams on Sunday if they can beat already-relegated Hearts.