Family members identified the man as Christian Acosta, whom Gonzalez said was 39 years old.
Sheriff's deputies said it began with a phone call from the family member of his ex-girlfriend shortly before 6 a.m.
The relative reported that the suspect kidnapped the 29-year-old woman and the toddler from a home in the 9400 block of Desert Flower near Highway 249 in the Fallbrook area.
Acosta was the child's father, and authorities said he and the woman were in a common-law relationship.
The man broke down the door, then fired shots before taking off with the pair, officials said.
By 9:15 a.m., the car they were in was spotted in Fort Bend County, and the chase started when the suspect refused to stop. Officers would be in pursuit for the next 45 minutes from Fort Bend through Harris and eventually to Chambers County. The chase reached speeds of 100 mph. An investigator and the woman's stepfather were on the phone trying to convince him to stop.
"I tried my best. I stayed 40 minutes on the phone with him," Arturo Escobedo said.
Officials said the man continued to refuse to surrender, adding that he spoke of "suicide by police."
Authorities ended up barricading the area of Plantation Drive at Highway 565 to block the suspect from reaching his house in the Cove area of Chambers County.
"We did not want him to get to the house because there were two more children he was coming to get," Chambers County Sheriff Brian Hawthorne said.
Once deputies stopped him, he reached for his gun, and that's when two snipers with Chambers County's Special Response Team shot and killed him, Gonzalez said.
The woman and child were unharmed. No members of law enforcement were injured either.
Watch: SkyEye coverage as scene unfolded in Chambers County
Suspect shot to death after holding ex-wife and toddler at gunpoint in car, officials say
According to Gonzalez, there was a history of abuse in the relationship, noting that the suspect had previously threatened to take the woman to Mexico and kill her.
The couple had recently split, and she stayed with relatives to escape him.
"It shows they will hunt you down and find you if they are hell-bent on doing that. Luckily, she didn't die," Gonzalez said.
It's believed no shots were fired during the chase.
You can reach the National Domestic Violence hotline 24/7 at 1-800-799-SAFE. You can also chat with someone live or text "START" to 88788.
The Houston Area Women's Center also has a 24/7 hotline available by calling 713-528-2121 or chat on their website.