The 39-year-old woman was temporarily living with her brother-in-law and his young son at their home in Livingston when the attacker sneaked into the detached house early on Saturday.
The attacker ordered the terrified man and his son at knifepoint into a room before subjecting his victim to the serious sexual assault.
The woman had only been staying at the house because she was in the middle of moving home.
Lothian and Borders Police confirmed that the attacker had tried to gain entry to up to four homes nearby in the hour before eventually finding an unlocked door at the address where the attack took place.
Police have made a fresh appeal for help in catching the attacker, who struck at around 5am at the property in Oakwood Park, in the Deans area of Livingston.
Officers who specialise in tracking down sex offenders have reportedly been drafted in, with the squad also examining UK-wide records in a bid to establish whether the attacker has struck before.
He is described as 6ft, in his 20s, with short dark hair, "chunky" sideburns, and wearing a cream hoodie top with the word "Crosshatch" across the chest, and a large pouch at the front. He also wore a blue baseball cap and jogging bottoms.
Detective Inspector Alan Somerville, who is leading the investigation, said: "We believe that our suspect has been trying doors, and knocking doors, in the hour before. I know of three, and possibly four (addresses]. I do not think that he targeted a particular house.
"The attack has left the victim seriously traumatised, and she is being supported by family and trained officers."
He added: "It is the brother-in-law and his child. The victim has been staying there between house moves."
DI Somerville said some residents looked out of their window and saw a man whose description was consistent with the attacker. Residents in the area are being urged to keep their doors and windows locked.
DI Somerville said: "In addition, we are renewing our appeal for any information that can help us trace the suspect."
It is understood that detectives are studying a number of CCTV security tapes taken from the homes of neighbours.
One neighbour, who lives very close to the house, said: "It is hard to believe something like that could happen here."
Another neighbour added: "I've lived here for five years and always left the door open. We are not leaving the door open now."