Published Sunday, 08 January 2012
They were asleep in the house in the Colinvale area of Poleglass when the device was pushed through the letterbox in the early hours on Sunday.
The child had to be treated for the effects of breathing in smoke but did not need hospital treatment, while the woman was uninjured.
Smoke damage was caused to the front of the house as flames quickly engulfed the front door and hallway.
Sinn Féin MLA for the area Jennifer McCann said neighbours tried to help.
"It seems a number of people came in the middle of the night and put some sort of flammable liquid into the doorway and set it alight," Ms McCann said.
"Only that a neighbour saw it and was able to put it out, this house would have gone up in flames and burnt to death the young child and mother who live there."
It is not clear why the family were targeted.
The SDLP's Alex Attwood has expressed disgust at the attack.
"Attacks like these, which deliberately endanger the lives of young children, are totally intolerable and must stop," the MLA said.
"It is extremely fortunate that no one has been killed here, but that is no thanks to the reckless, callous and cowardly actions of those responsible."
Police investigating the incident want to speak to a man who got into a taxi shortly after the incident.
They have also asked any taxi drivers who were in the Colinvale area at around 2.30am to get in touch.
Information can be passed on to Lisburn CID on 0845 600 8000.