The rescue workers and local villagers at the accident site were digging for survivors from under heavy boulders and mounds of earth.
The incident happened at night in Bilaspur, about 60 miles (100 kilometres) from Shimla, the capital of the picturesque Himalayan state.
"15 deaths have been confirmed so far," the local government said in a statement.
State Chief Minister Sukhvinder Singh Sukhu "expressed deep sorrow and anguish over the tragic bus mishap," the statement added.
"Three children have been rescued alive and were being treated" at a local hospital, and the rescue operations were ongoing.
It is the latest such incident in the world's most populous country, where flash floods and landslides have already claimed dozens of casualties this monsoon season.
Monsoon rains, usually from June to September, bring respite after peak summer months and are also essential for the region's agrarian economy.
But they also bring widespread death and destruction, particularly in ecologically fragile Himalayan states, where experts warn that the number of extreme events has increased in recent years.
Torrential rains last week also triggered deadly landslides and floods in northeastern India's Darjeeling, destroying swathes of premier tea estates in the Himalayan region.
The deluge, which killed at least 36 people in the region, wiped out around five percent of Darjeeling's renowned tea gardens, delivering a heavy blow in a district that has become synonymous with the leaf itself.
The disaster also washed away many roads and destroyed more than 500 houses.
Hundreds of locals and tourists also moved to safe shelters, waiting for rescue or waters to recede from key arterial roads.
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