The family, named locally as Peter and Tracey Wilkinson and their son, Pierce, were found with knife wounds in Stourbridge at about 08:00 BST.
A Land Rover taken from the house was intercepted by West Midlands Police about 30 minutes later in Norton Road.
A man in his 20s has been arrested on suspicion of murder and wounding.
The family were found after emergency crews were called to Greyhound Lane, Norton.
Police said they understood the suspect was known to the victims and may have been living at the address, although he was not a family relative.
The 13-year-old boy and his mother were found in the house with critical stab wounds.
The woman was pronounced dead at the scene, while the teenager was later pronounced dead in hospital.
Mr Wilkinson, believed to be a company director for local firm Hill & Smith Ltd, was found in the garden of the house.
He remains in hospital with stab wounds to his chest and back.
Get news from the BBC in your inbox, each weekday morning
Peter Wilson, BBC Midlands Today special correspondent, said he understood the couple also have an older daughter who is attending university.
Redhill School in Stourbridge confirmed the boy was a Year 8 pupil.
In a statement, the school said pupils and staff were being supported and added: "Our thoughts and prayers are with those who have been affected by this."
Police said that as the Land Rover was intercepted, it was in collision with a police car.
The force said two officers were taken to Russells Hall Hospital, in Dudley, with minor injuries. They have since been released.
Supt Lee Kendrick, of Dudley Police, said: "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 that "tragically" there was "nothing that could be done" to save the woman, understood to be in her 50s.
Supt Kendrick said they "don't think this was a burglary or a robbery".
The man in his 20s would be questioned on suspicion of murder and wounding, the force said.
Police are not looking for anyone else in connection with the attack.
A mother and her 13-year-old son have been killed and the woman's husband seriously injured in a stabbing attack at their home.