Fire crews attended the blaze at Bute Tower, in Clifton Terrace, and found the building engulfed in smoke.
The alarm was raised at around 9pm on Saturday, after the fire took hold in a bin chute.
Two women and a man were found to be suffering from smoke inhalation and taken to Hairmyres Hospital.
Fire chiefs said the incident is not being treated as suspicious.
Station Commander Robert Deans said: "By giving an early warning of the danger, working smoke alarms saved lives."