Police said Haji Mohamed, 8, was struck by the school bus when he attempted to run across Sprague Street.
Road accidents are common in India, often due to poor driving or badly maintained roads and vehicles.
The adults killed included the bus driver, two teachers and another woman, officials said.
No other injuries were reported at the scene, he said.
Police say it was carrying about 40 people, a lot of them students at a private school who were on their way home.
Prime Minister Narendra Modi said he was anguished by the loss of lives in the latest incident.
"My prayers and solidarity with those who lost their near and dear ones".
The Himachal Pradesh government has ordered a team of doctors to remain on standby.
An injured student, Ranveer Singh, told the Indian media that he fell out of a window after the bus started rolling into the gorge. Most of those killed were students of elementary classes, aged below ten.
Initial reports indicated the bus driver was speeding as he took the children home and lost control at the edge of the gorge.
The students are from Wazir Ram Singh Pathania Memorial Public School. Numerous students in the bus were in Class 5 and below, IANS reported.
According to news agency ANI, Himachal Pradesh Chief Minister Jairam Thakur has announced Rs 5 lakh each for the family of the accident victims.
A rescue operation to look for potential survivors among the kids - most of whom were under the age of 10 - will continue through Monday night, Kumar said. "I have spoken with the Chief Secretary, Director General and Deputy Commissioner".
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