Two students were injured in the murder-suicide at North Park Elementary School in San Bernardino, and one of those children later died at a hospital, Burguan said.
Burguan said the shooter, Cedric Anderson, 53, of Riverside, walked into the classroom armed with a large caliber revolver and opened fire "without saying anything," killing Karen Elaine Smith, 53, also of Riverside. Smith was a teacher at the school.
The two students struck by gunfire were standing behind Smith, the police chief said.
Jonathan Martinez, 8, died at Loma Linda University Medical Center after being airlifted from the school, the police chief said. The other student, a 9-year-old boy, was listed in stable condition at the medical center, according to the hospital.
Police do not believe the two children were targeted; they happened to be near the teacher.
Burguan said that at the time of the shooting there were 15 students from grades one through four in the special needs classroom, along with two adult aides.
"This is a tragic incident that has befallen our city," San Bernardino Police Lt. Michael Madden told reporters.
How shooting unfolded
Police received reports of the shooting at the school shortly before 10:30 a.m., and officers responded within minutes, Madden said.
San Bernardino Police Capt. Ron Maass, the incident commander, said Anderson had gone to the classroom to visit Smith. There is no indication the suspect's handgun was visible when he entered the school, Maass said.
Anderson signed in at the front office when he entered the school, according to Maass. He said he believed Anderson was known to the front office.
Burguan said Anderson told employees in the office he wanted to drop something off with his wife and then headed to the classroom. He added that it is not uncommon for a spouse to gain access to a school campus to meet with their partner.
"He came in, and very, very quickly upon entering the classroom started shooting," Burguan said.
Anderson also reloaded after firing what is believed to have been a .357-caliber revolver, the police chief said.
Burguan said preliminary information indicates the two were recently married.
"I'm told that their marriage was relatively short. They've only been married for a few months and they've been separated for about the last month, month-and-a-half, roughly, when this incident took place," he said. "But there's nobody that in the investigation has come forward to say that they saw this coming."
The police chief said Anderson "does have a criminal history."
Court records showed that he had faced criminal charges of brandishing a weapon, assault and crimes against public peace in 2013, with those charges later "dismissed or not prosecuted." There had also been two petitions for temporary restraining orders filed against him by women in previous years.
Another tragedy for San Bernardino
Earlier in the day, San Bernardino Mayor R. Carey Davis said: "Far too often in our country and throughout the world, we gather to report this news of tragic events that take place."
San Bernardino was the site of a mass shooting
in December 2015, in which 14 people were killed when a couple opened fire at a holiday party held by the county's health department.
The mayor said that after the 2015 terror attack, the community came together and showed that it could be "very resilient and can deal with tragedy, disaster just as effectively as any community in the United States. And we did that. And we will continue to do that in this occasion."
"We will rise to help our parents, our teachers to regroup and to once again gain confidence in our community," Davis said.
Frantic parents seek to reunite with their kids
On Monday, North Park students were evacuated and reunited with their parents at nearby Cajon High School, police said. Video showed community members lining up to cheer on and give high-fives to students arriving at Cajon High.
Frantic parents rushed to North Park Elementary after the shooting.
Brad Hendran told CNN affiliate KABC
he was looking for his daughter, who is in the second grade.
"I fell to my knees and started saying Our Fathers and Hail Marys. ... I'm gonna hug her. ... I'm gonna hug her. I hope she is OK," Hendran said.
One father, who wasn't identified by the CNN affiliate, stood behind police tape, looking for one of his sons. He said his two boys are in second grade and fifth grade.
"I see one of my boys, but not the smallest one," he said tearfully. His eldest son had just gotten on a bus, he said.
He was at work when he heard about the shooting from the boys' mom, who works for the school district.
North Park Elementary will be closed for at least the next two days.
"I would like our community and our nation to keep our families in prayer," San Bernardino City Unified School District Superintendent Dale Marsden said.
"It's going to take time for our heads, our hearts to heal." Marsden added.
A prayer vigil was held Monday night at a San Bernardino church, Our Lady of the Assumption.