The carnage began with the 18-year-old suspect, identified as Salvador Ramos, shooting his own grandmother, who survived, authorities said.
He fled that scene and crashed his car near the Robb Elementary School in Uvalde, Texas, a town about 130km west of San Antonio. There he launched a bloody rampage that ended when he was killed, apparently shot by police.
The motive was not immediately clear.
Law enforcement officers saw the gunman, clad in body armour, emerge from his crashed vehicle carrying a rifle and "engaged" the suspect, who nevertheless managed to charge into the school and open fire, Texas Department of Public Safety (DPS) Sergeant Erick Estrada said on CNN.
Speaking from the White House hours later, a visibly shaken US President Joe Biden urged Americans to stand up to the politically powerful US gun lobby, which he blamed for blocking enactment of tougher "common-sense" firearms safety laws.
Biden ordered flags flown at half-staff daily until sunset on Saturday in observance of the tragedy.
Governor Greg Abbott said that the suspect, identified as Salvador Ramos, was apparently killed by police officers, and that two officers were struck by gunfire, though the governor said their injuries were not serious.
Authorities said the suspect acted alone.
After conflicting early accounts of the death toll, Texas public safety officials said that 19 school children and two teachers had died.
The school's student body consists of children in the second, third and fourth grades, according to Pete Arredondo, chief of the Uvalde Consolidated Independent School District Police Department, who also addressed reporters. Pupils in those grades would likely have ranged in age from 7 to 10.
The carnage unfolded 10 days after 10 people were killed in Buffalo, New York, in a predominantly Black neighbourhood. Authorities have charged an 18-year-old man who they said had travelled hundreds of miles to Buffalo and opened fire with an assault-style rifle at a grocery store.
Tuesday's bloodshed in Texas began when the suspect shot his grandmother before going to the school, Texas Department of Public Safety officer Chris Olivarez said on Fox News, a development Abbott mentioned earlier in the day.
"I have no further information about the connection between those two shootings," the governor said.
University Hospital in San Antonio said on Twitter that it had received two patients from the shooting in Uvalde, a 66-year-old woman and a 10-year-old girl, both listed in critical condition.
Uvalde Memorial Hospital said 15 students from Robb Elementary were treated in its emergency room, with two transferred to San Antonio for further care, while a third patient transfer was pending. It was not immediately clear whether all of those students survived.
A 45-year-old victim grazed by a bullet was also hospitalized at Uvalde Memorial, the hospital said.
Hours after the shooting, police had cordoned off the school with yellow tape. Police cruisers and emergency vehicles were scattered around the perimeter of the school grounds. Uniformed personnel stood in small clusters, some in camouflage carrying semi-automatic weapons.
The Texas rampage capped a series of mass shootings in US schools that have shocked the world and fuelled a fierce debate between advocates of tighter gun controls and those who oppose any legislation that could compromise the right of Americans to bear arms.
Tuesday's shooting is one of the deadliest at a US school since a gunman killed 26 people, including 20 children from 5- to 10-years old, in a rampage at Sandy Hook Elementary School in Connecticut in December 2012.
In 2018, a former student at Marjory Stoneman Douglas High School in Parkland, Florida, killed 17 students and educators.
Another crisis for Biden
US President Joe Biden, who ordered flags flown at half-staff until sunset daily until May 28 in observance of the tragedy, was to address the nation later on Tuesday, the White House said.
Biden spoke with Governor Abbott from Air Force One and offered him "any and all assistance" needed.
"Enough is enough," Vice President Kamala Harris said before Biden spoke. "As a nation, we have to have the courage to take action and understand the nexus between what makes for reasonable and sensible public policy to ensure something like this never happens again."
The latest shooting hands Biden, already facing the lowest approval ratings of his presidency, another crisis on top of 40-year high inflation rates and the war in Ukraine.
It came just 10 days after a mass shooting in a Buffalo, New York, grocery store, and piled pressure on his administration to make good its vows to crack down on gun violence.
When he ran for the presidency, Biden promised to push gun safety measures and reduce the country's tens of thousands of annual gun deaths. Biden and his fellow Democrats have failed to get enough votes in Congress for background checks for gun purchases or other proposed bills.
The United States is the most heavily armed society in the world, according to the Geneva-based Small Arms Survey, a research group.
Small, rural states where gun ownership is widespread have disproportionate influence in the Senate, where a supermajority of 60 votes is needed to advance most legislation in the 100-seat chamber.
The US experienced 61 "active shooter" incidents last year, up sharply from the prior year and the highest tally in over 20 years, the FBI reported this week.
