Police said an ambulance speeding towards the emergency wing lost control before hitting a rickshaw and pedestrians.
The dead has been identified as Goleynur Begum, 25 and her 7-year-old son 'Sakib'.
The third victim, estimated to be 45 years old, is yet to be identified.
Begum's husband Ferdous told bdnews24.com that they came from Patuakhali two days ago to treat his son Sakib for injuries sustained in a road accident.
His other son, 6 months old 'Akash,' was with his mother but suffered injuries on the head.
Sakib died on the spot while his mother succumbed to her injuries around 3pm.
DMCH police camp's Sub-Inspector Bachchu Mia said the speeding ambulance, with only the driver inside it, entered the DMCH premises and rammed into a rickshaw before hitting the people.
"The mid-aged man died on the spot and the child was declared dead by doctors at the emergency department," he told bdnews24.com.
Five others injured have been admitted to the DMCH. Of them, a six months pregnant woman Amena Begum lost her child, said doctors.
Police said the ambulance is owned by one Nasir Mistri, but he was not in the vehicle when the incident occurred.
"His assistant Sohel was behind the wheels. He has been detained," said Sub Inspector Mia.