The bus was travelling on a hilly stretch near Bilaspur district late on Tuesday when a landslide struck following days of torrential rain.
At least 20 to 25 passengers were on the bus at the time.
Nine men, four women and two children were among those killed, police said.
Three injured children were rescued and admitted to hospital for treatment, according to the office of Sukhvinder Singh Sukhu, the state's highest-elected official.
Rescue operations continued on Wednesday trying to find other missing passengers who are believed to be dead, police said.
Intermittent rains have lashed the region since Monday, making the fragile mountain slopes unstable.
President Draupadi Murmu and Prime Minister Narendra Modi offered their condolences following the deadly landslide.
Extreme rains have caused flooding and landslides across the South Asian region, which includes India, Pakistan, Bangladesh, Bhutan, Sri Lanka, Afghanistan, Maldives and Nepal.
Flash floods swept away an entire village in India's northern state of Uttarakhand in August, while at least 44 people were killed in neighbouring Nepal at the weekend due to mudslides and flooding triggered by severe rainfall.
The weekend's heavy rainfall arrived at the end of Nepal's monsoon season, which usually begins in June and ends by mid-September.
It also left parts of the capital, Kathmandu, flooded and caused the cancellation of all domestic flights on Saturday.
Experts say human-caused climate change is intensifying South Asia's monsoons, which traditionally run from June to September and again from October to December.
The rains, once predictable, now arrive in erratic bursts that dump extreme amounts of water in short periods, followed by dry spells.
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