The children were found about 48 miles away in Texas City.
Police responded to a welfare concern at an apartment in Texas City about 6:15 p.m. Thursday and found three children ages 5, 2, and an infant dead. A woman at the scene had a gunshot wound to the head and was taken to the hospital, where she's in stable condition.
Before the arrest, detectives had said they're looking for a suspect identified as Junaid Hashim Mehmood. They described him as "armed and dangerous," and warned people not to approach him if they see him.
Mehmood, 27, was arrested 25 miles away from the apartment complex where the victims were found. He was taken into custody on an unrelated warrant, Texas City Police said in a statement.
The cause of death for the three children has not yet been made available. The suspect's relationship to the children is unclear.