South Yorkshire Police said an accelerant was set alight after it was poured through the letterbox of a house in Montford Drive, Burngreave, at 12.40am on Thursday, May 25.
The fire caused damage to the front door and entrance hall of the property but the woman and children inside managed to escape.
A police probe into the arson attack is now underway.
Officers arrested a 21-year-old man on suspicion of arson and later released him while enquiries continue.
They want witnesses or anyone with information to come forward.
A South Yorkshire Police spokeswoman said: "Police investigating a suspected arson attack on a house in Sheffield are asking for witnesses to come forward.
"At about 12.40am on Thursday, May 25, an accelerant is said to have been poured through the letterbox of a house in Montford Drive, Burngreave, while the occupants were asleep upstairs.
"The fire caused damage to the front door and entrance hall of the house but thankfully no one was injured.
"Officers are asking anyone who saw or heard anything suspicious in the area in the early hours of that morning to get in touch.
"A 21-year-old man arrested on suspicion of arson has been released under investigation as enquiries continue."
A spokeswoman for South Yorkshire Fire and Rescue said the front door was well alight when crews arrived at the house.
The occupants had all managed to get out unharmed.
The spokeswoman added: “Starting fires is reckless and costs lives.
"In recent, high-profile, national cases, arson attacks on homes have resulted in murder convictions and substantial prison sentences.
“The maximum sentence for arson is life in prison and we’d encourage members of the public who know of people starting fires in their area to contact the police.”
Call South Yorkshire Police on 101 or Crimestoppers on 0800 555111.