David O'Brien got on the end of a David Graham cross to head Allan Moore's men into a 15th-minute lead.
The visitors were the better side throughout the first half while Billy Reid's outfit struggled to find any fluency, causing their frustrated fans to voice their displeasure.
Their misery was compounded when Accies' captain, Alex Neil, was sent off in the third minute of first-half stoppage time for a second bookable offence.
At this stage, Ton looked certainties to pick up the three points, but this feeling of comfort was perhaps what caused their late downfall.
They dominated possession for much of the second period but played much of it at walking pace and without looking to press home their one-man advantage by going for the jugular.
And their failure to find a killer second was punished in 81 minutes when Ali Crawford drilled home from 10 yards after the Ton defence failed to clear their lines.
Stand-in Morton skipper Michael Tidser was shown a straight red card for foul and abusive language just five minutes later to put the seal on what had turned out to be desperately disappointing day for the away side.
Morton manager Allan Moore said: "I'm very disappointed to say the least. We warned the boys to keep the pressure up in the second half and that's exactly what we didn't do."
Morton (4-5-1): Gaston; Taggart, Reid (O'Ware, 50), Mensing [Trialist], Dyer; Graham, Stirling, Wallace (Bachirou [Trialist], 66), Stirling, Tidser ©, O'Brien; Weatherson (Campbell, 75).
Subs not used: Wilkie, Combe (gk).
Sent-off: Tidser (86)
Hamilton (4-1-3-2): Cuthbert; McGlinchey (Gillespie, 66), Canning (Gordon, 23), Devlin, Hendrie; Neil; Routledge, Fisher (Fraser, 68),Crawford; McShane, Longridge.
Subs not used: MacKinnon, Allan (gk).
Booked: Neil (34)
Sent-off: Neil (45+3)
Referee: Crawford Allan