A woman was also fighting for her life after suffering a gunshot wound to the head during the incident in Spring, outside Houston, Harris County Constable Ron Hickman told AFP.
The woman was airlifted to an area hospital, where she remains in critical condition, according to the Harris County Sheriff's Office.
One of the killed children died after arriving at hospital. The two slain adults were men, Hickman said.
Before being taken to hospital, the woman was able to tell authorities where the suspect, a relative, was headed, triggering a police chase that lasted at least 20 minutes.
"We got him in a cul-de-sac," said Hickman, after at least an hour had passed during which the suspect sat in his vehicle surrounded by about 50 police with their guns drawn. There were at least a dozen patrol cars waiting to pounce.
"Members of the Harris County Sheriff's Office High Risk Operations Unit and Hostage Negotiation Team are currently negotiating the surrender of one armed suspect," the Harris County Sheriff's Office said in a statement.
The suspect had not taken any one hostage.
Law enforcement officers surround a shooting suspect in his car Wednesday, July 9, 2014, in Spring, Texas. -AP Photo/Houston Chronicle, Brett Coomer
Hickman said a large military-style SWAT truck was parked right in front of the suspect's vehicle to prevent him from fleeing."The suspect is engaged in a dialogue with our sheriff's office," Hickman said.
Some residents took to Twitter to thank the authorities.
The shooting was the latest in a spate of similar incidents in a country plagued by gun violence.
President Barack Obama has warned that America needs to do some "soul-searching" on gun control, which remains relatively lax despite regular gun rampages.
On May 23, a student with mental problems killed six people and then himself in California, while on June 5 a gunman killed one person and injured two others on a campus in Seattle.
In June, a couple with possible anti-government militia links shot dead two police and a civilian in Las Vegas, and a teenage gunman shot a 14-year-old student dead at an Oregon high school.
