The visitors were a lick of paint away from taking the lead in the second minute as Grant Gillespie found space in the box and crashed an effort against the post.
Hamilton continued to press their opponents and dominate possession, but they suffered a major setback early in the second half when Dougie Imrie was shown a red card for violent conduct.
Dundee tried to make their numerical advantage count and almost broke the deadlock on the hour mark as Stewart hit one from just outside the area, with his effort narrowly missing the top corner.
Canning must have thought that he had secured his first win in charge of the Accies when he nodded home Ali Crawford's cross five minutes from the death, but there was more late drama to come.
With just three minutes remaining, Stewart rifled one into the top corner from outside of the box to rescue a point for the Dark Blues with the last meaningful chance of the afternoon.