When crews arrived at the house in Sherwood Road they found the family already outside. A woman in her 20s was suffering from a possible broken ankle and was taken to hospital, also suffering from the effects of breathing in smoke, along with the baby and a man in his 30s.
Four other people were also checked over by medics for smoke inhalation. Two dogs escaped the fire unharmed.
Firefighters wearing breathing apparatus entered the house and found a fire in the kitchen which they dealt with using a high pressure hose reel and a jet. Ventilation fans were used to clear the smoke.
An investigation into the cause of the fire found it started accidentally after a lamp left on overheated and caught light. The smoke alarm fitted in the property had not activated.
An Avon Fire & Rescue Service spokesman said: “We would like to remind people to ensure they turn off all electrical items which don’t need to be left on continuously overnight.
“Smoke alarms should be fitted on every level of the home and the batteries should be checked at least once a week.”