The girl was shot in the head following a heated argument, the police said. She is a class 10 student at a Delhi school.
They were rushed to a nearby hospital after the incident.
The girl's condition was stated to be critical, while the boy was recuperating.
An FIR has been lodged against the class 11 boy following a complaint by the girl's uncle, Assistant Superintendent of Police Anoop Singh said.
The boy's father has been summoned for questioning and the licensed pistol has been recovered as well by the police.
The incident took place on Wednesday evening when they were returning home from a coaching centre on a scooty, police said.
(With inputs from PTI)