The storm toppled trees and caused houses to collapse in Rajasthan and Uttar Pradesh.
Relief commissioner Sanjay Kumar said the devastation is particularly severe in Agra, the northern Indian city where the Taj Mahal is located.
He said 36 people have died there on Wednesday night as the wind speed touched 130km (80 miles) per hour.
Some people also were killed by lightning.
The Press Trust of India news agency said at least 27 others have died and another 100 are injured in Rajasthan.
Vasundhara Raje, the chief minister of the state, said officials were heading to the affected areas to begin relief work and restore utilities.
"An unfortunate incident, we have been working closely with local authorities to mitigate the situation," she said on Twitter.
Have directed Ministers and concerned officials to affected areas to begin relief work immediately and restore utilities. An unfortunate incident, we have been working closely with local authorities to mitigate the situation.
Narendra Modi, the prime minister of India, said he was “saddened by the loss of lives due to dust storms in various parts of India.
“Condolences to the bereaved families. May the injured recover soon.”
Mr Modi said he had “asked officials to coordinate with the respective state governments and work towards assisting those who have been affected.”
Saddened by the loss of lives due to dust storms in various parts of India. Condolences to the bereaved families. May the injured recover soon.
The storm caught people by surprise as the monsoon season is still more than six weeks away.
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