Several bursts of gunfire erupted about 6:10 p.m. near a Rainbow Foods store in the eastern Twin Cities suburb, KMSP-TV reported.
The gunman fired at a car carrying the 10-year-old and the child's mother, Oakdale Police Chief Bill Sullivan said. The child died and the mother was wounded in the arm, said Sullivan, who would not say whether the child was a boy or a girl.
A woman in another car was struck and wounded in the hand, Sullivan said. Both women are expected to recover, he said.
"It appears to be entirely random," Sullivan told The Associated Press. "At this time it appears he was randomly shooting at vehicles."
A 33-year-old man was in custody and was being questioned by investigators, Sullivan said. The St. Paul Pioneer Press reported the man lives in the neighborhood.
Authorities believe there are no other suspects and there is no danger to the public, Sullivan said.
"It's sick," Oakdale Mayor Carmen Sarrack told the Minneapolis Star-Tribune on Monday night. "What can you say when a 9-year-old or 10-year-old is killed? ... People are devastated that this could happen in our city."