Ronald Lee Haskell, 33, has been charged with multiple counts of capital murder after the rampage in the middle-class Houston suburb of Spring during which a 15-year-old girl was also wounded.
Harris County Sheriff's Deputy Thomas Gilliland said after a brief chase, the man held deputies at bay for three hours before he surrendered.
Mr Gilliland said the dead children were two boys, aged four and 14, and two girls, aged seven and nine and their parents, a 39-year-old man and a 33-year-old woman.
The gunman, who gained access to the house pretending to be a FedEx delivery man, was estranged from a woman related to the couple. Police said he was wearing a FedEx uniform though it is not yet known if he worked for the company.
The injured girl contacted authorities telling them the gunman was en route to shoot other family members.
The sheriff's department said officers were called to the house at around 6pm on Wednesday and found two adults and three children dead. Another child later died in hospital.
Assistant Chief Deputy Constable Mark Herman said: "It appears this stems from a domestic issue with a breakup in the family."