Another disaster-management official confirmed the death toll, saying the stampede took place at about 2:45am (2115 GMT).
"A high-level inquiry has been ordered by the government into the stampede incident at the Shri Mata Vaishno Devi Shrine," he said.
"Extremely saddened by the loss of lives," prime minister Narendra Modi said on Twitter, adding that he was in touch with local officials.
"Condolences to the bereaved families. May the injured recover soon," Modi said.
Other senior officials including the Indian president offered their condolences, and the home minister called the news "heart-wrenching".
Government minister Jitendra Singh said he was on his way to the site to take stock of the situation.
Reports cited witnesses as saying that the daily limit of 25,000 was exceeded and that large groups entered without permission slips.
Other reports suggested an argument had broken out between devotees.
Rescue operations started immediately and the injured -- some of whom were reported to be in a serious condition -- were taken to hospital.
Video footage on social media showed small ambulances with flashing lights rushing to hospitals while it was still dark.
The shrine, open 24 hours a day, is located in the Katra hills some 30 kilometres from the city of Jammu.