Most of the injuries were minor, and none was believed to be life-threatening, Missouri Highway Patrol Sgt. Al Nothum told CNN affiliate KTVI-TV.
One child was ejected out the back emergency door but appeared to have suffered little more than sprains and cuts and was talking while being taken to a hospital, Nothum said.
Emergency workers helped the trapped driver out of the bus, Nothum said. The driver suffered some cuts at minimum, he said.
"The bus driver, we've got to give her credit ... she kept that bus upright. ... She did some good driving here," Nothum told KTVI.
Aerial video from KTVI showed adults comforting some of the children, who had been bound for Hanna Woods Elementary School in Ballwin, Missouri, near a guardrail just feet from the crash site. Emergency vehicles blocked two of the four westbound lanes, slowing traffic for part of the morning.
The bus was heading west on I-44 near the suburbs of Kirkwood and Crestwood when a car ahead of it struck a median wall while its driver tried to pass a truck, Nothum said, citing witnesses.
The car spun toward the bus, and the bus driver swerved to avoid it. The bus then went through a guardrail and down an embankment and into a ditch with trees and shrubs, said Nothum, who added that most of the children left the bus on their own.
The driver and at least 12 children were taken to hospitals, Notham said. Kirkwood Fire Chief Jim Silvernail said 13 children were injured.
No traffic citations were immediately announced. Nothum said investigators were asking the car driver for his or her account of what happened Thursday morning.
The children's ages weren't released. The school serves children in grades K-5, Parkway School District spokesman Paul Tandy said.