Police said the 44-year-old man, from Manchester, is believed to be known to the victims.
Greater Manchester Police officers were called by colleagues from North West Ambulance Service at about 9.30pm on Thursday following the domestic incident in Miles Platting.
The 15-year-old boy was treated at the scene for stab wounds before being taken to hospital where he died around an hour later, the force said.
Police said his mother, in her 40s, was treated at the scene for serious stab wounds and is now in a stable condition in hospital.
Detective chief inspector Alicia Smith said: “Last night’s attack has left a community reeling and a family coming to terms with what was an absolutely devastating incident for everyone involved.
“A teenage boy who had his whole life ahead of him has tragically lost his life and not only is his mother grieving at the loss of her son, but she is also recovering from what will have been no doubt, hugely traumatic.
“Our thoughts remain very much with the family and we are absolutely committed to establishing what exactly happened last night.
“Detectives have been working around the clock since this incident was first reported to us and we’ve now made an arrest which is a huge step forward in getting the family the answers they need.
“Our inquiries are ongoing and we would still like to hear from anyone with information, or anyone who may have seen anything. Report it online or via LiveChat at www.gmp.police.uk or alternatively, contact Crimestoppers anonymously on 0800 555 111.”