The family were able to escape from the property in Poplar Avenue when the smoke alarm sounded at 9.10pm on Friday.
The cigarette was in a plastic ashtray on top of a pillow which caught light and spread through the bedroom.
Watch manager Glyn McGann, of Horwich fire station, said the results could have been “catastrophic” if it was not for the smoke alarm.
The couple, aged 48 and 50, and their 24-year-old daughter were checked over at the scene but did not need hospital treatment.
Mr McGann said: “There was some smoke damage to the bedroom and adjacent bedroom.
“They had a smoke alarm upstairs which alerted them to the fire. If it was downstairs it might not have done.
“This could have been catastrophic. The message is to keep smoke alarms in working order and make sure cigarettes are fully extinguished and put in a non- combustible container.”
Two new smoke alarms were fitted at the house.