Indian Home Minister Sushilkumar Shinde on Saturday said up to 73,000 people had been rescued so far from various sites within the state.
In the adjacent state of Himachal Pradesh, 20 people have been killed and around 1,200 tourists remained stranded in remote rain-hit regions.
The Kinnaur district has seen heavy damage with power cut off to large areas and roads washed away.
The monsoon, which covers the subcontinent from June to September, usually brings some flooding. However, the heavy rains arrived early this year, catching many by surprise and exposing a lack of preparedness.
Floods and landslides from monsoon rains have also struck Nepal, leaving at least 39 people dead, according to the government in Kathmandu.