Thomas O'Ware fired David Hopkin's men in front on seven minutes when he side-footed home from six yards after Colin McMenamin had flicked an Aidan Fulton corner into his path.
Kyle Wilkie, pictured, was Morton's standout performer on the night, and he rifled a crisp left-footed strike into the bottom right corner on 26 minutes to double his side's advantage.
Despite creating several openings, two of which Peter Weatherson should have converted, the hosts failed to put the match to bed and were duly punished as they collapsed in the final 30 minutes.
Daniel Hepburn threw Stenny a lifeline with a stunning strike on the hour and sub Jack Hamilton completed the comeback by curling a fantastic free-kick past David Hutton nine minutes from time.
The draw leaves Morton a point behind leaders Rangers with one game left to play.
Ton face Partick Thistle, with date and venue to be confirmed, while the Ibrox side are at Queen's Park next Monday.