The bodies of the woman and child were discovered on Wednesday after hikers found a 5-year-old boy who said his father had taken the family to a cave and attacked them.
Early reports have suggested that the mother and eldest son were beaten to death, although the exact cause of death is not yet known.
The father responded violently to the arrest and refused to reveal the location of the rest of his family members, according to sources from El Pais newspaper.
It is believed the mother and two children, who are all German nationals, had travelled to Tenerife on Monday to visit their father, who is also German and reportedly lives on the island as a cook.
The hikers discovered the boy covered in dirt and crying near the town of Adeje, Tenerife, according to El Pais.
The boy, who did not speak Spanish, told a translator he had escaped from the cave after he saw a large amount of blood.
More than 100 emergency workers and polic were called to help search caves in the area.
The surviving child will now be looked after by his maternal grandparents, who flew out to Tenerife this week to take care of him.
Local officials are treating the deaths as a domestic violence case and a local court that specialises in crimes against women has taken charge of the investigation.
In a statement, he said: “The worst news arrives from Adeje (Tenerife). A woman and her son have been killed. Again the #ViolenciaMachista [gendered violence] hits double. A son of the couple managed to escape. All my love for him, for his family and loved ones. We have to end this.”
Violence against women is a persistent problem in Spain, which saw its worst year for gendered violence in 2017, and Mr Sanchez has made women’s rights a key part of his campaign in the country’s general election this month.
More than 990 women have been killed by their partners or former partners since 2003, when official Spanish records on such killings began.
Agencies contributed to this report