Two young children and their mother fled from their Merseyside home after inflammable liquid was poured through their door and set alight.
They were treated for smoke inhalation after their door's window was smashed and the house on Queens Road, Crosby, set on fire at 2200 BST on Monday.
Police said two teenagers were seen running from the scene towards Northern Road.
The family, who were all in bed, were "lucky to escape", police said.
The suspected arsonists were dressed in black clothing and one them had flames coming from his sleeves, police said.
Det Insp Dave Keegan said: "Those living in the house were all in bed at the time of fire and were lucky to escape without being really injured, this has clearly been a terrifying ordeal for them.
"Witnesses have said that one of the men running from the scene appeared to be on fire, it is very possible that he has received injuries as a result.
"I would appeal to members of the public who know somebody with fresh burns to contact us."