They and the woman’s husband were discovered at the house in Stourbridge shortly after 8am on Thursday.
A West Midlands Police spokesman said the woman, aged in her 50s, was pronounced dead at the scene and the teenager was pronounced dead in hospital.
The husband remains in hospital with stab wounds to his chest and back.
Detectives said a man aged in his 20s was arrested in a nearby road 30 minutes later in a Land Rover stolen from the scene in Greyhound Lane.
Superintendent Lee Kendrick, said: “Clearly our investigation is at a very early stage but we understand the suspect was known to the family and may have been living at the address - although he is not a family relative.
“We don’t think this was a burglary or a robbery.”
Mr Kendrick added: “What has unfolded here this morning is horrific and heart-breaking. Police and ambulance were confronted with an awful scene: a man, woman and boy all with very serious stab injuries
“The woman and boy were found in the house and the man in the garden.”
He added: “Our investigation is progressing at pace and we are working to understand exactly what the relationship was between the man and the family.
“At this stage we are not looking for anyone else in connection with the incident.”
A West Midlands Ambulance Service spokesman said: “Unfortunately, despite the best efforts of ambulance staff it became apparent at the scene that nothing could be done to save the woman and she was confirmed deceased.
“The teenage boy was found to be in cardiac arrest and received specialist trauma care and advanced life support from our staff.
“He was taken on blue lights to Birmingham Children’s Hospital with treatment continuing en route.
“Sadly, shortly after arrival in A&E and despite the best efforts of ambulance staff and hospital medics, he could not be saved and he was also confirmed dead.
“The man, who was conscious, was also in a serious condition as a result of his stab wounds and was alerted in to Queen Elizabeth Hospital, Birmingham by land ambulance.”