Firefighters were called to a communal bin room at Grampian Court shortly before 23:00 on Monday.
In addition to the baby, two men and a woman were taken to hospital. They were not seriously hurt.
A further 17 residents were checked at the scene by paramedics before being allowed to return to their homes.
Police Scotland said they were investigating a second wilful fireraising also in the Torry area of the city.
Emergency services were called to the fire outside Tesco on Wellington Road at about 22:15. The fire was extinguished and there were no injuries.
Police said officers were particularly keen to trace a group of six people who were seen in the area of Wellington Road prior to the first fire.
Insp David Paterson, of the Torry Community Policing Team, said: "Fire creates a significant danger to both the emergency services and the people in the area.
"With the latter of these two incidents, we've seen four people taken to hospital and we can be thankful that things did not take a turn for the worse.
"We are pursuing a number of lines of enquiry and can reassure the local people we will carry out a robust investigation"
Scottish Fire and Rescue Service group manager Fraser Burr said: "The service would like to remind people that those responsible for setting deliberate fires not only place our attending fire crews at risk but also potentially impact on our ability to respond to more important emergencies elsewhere in the city."
A three-month-old baby boy has been treated for the the effects of breathing in smoke after a wilful fire in an Aberdeen block of flats.