Madhya Pradesh's Women and Child Development Minister Archana Chitnis has ordered an inquiry into the incident.
Sinha, a former national-level volleyball player, had yesterday said she felt "greater pain" while visiting the temple than climbing Mount Everest, the world's highest peak at 8,844 metres.
The temple authorities said men had to be clad in "dhotis" (shola), while women had to wear a saree to enter the sanctum sanctorum during the "bhasma aarti", a unique practice observed at the temple in the early hours every day.
He added that Sinha had come to the temple at around 4.30 am on December 24 and was told about the tradition by the priests and security personnel there.
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Sharma said generally, those not wearing a dhoti or a saree were allowed to enter the sanctum sanctorum, but the entry was restricted during the "bhasma aarti".
He said entry to the sanctum sanctorum during the "bhasma aarti" was only allowed through pre-booked slots and added that Sinha had reached the temple on a short notice.
However, Sinha told reporters that the footage also showed a man entering the temple's sanctum sanctorum wearing jeans.
"Not allowing a person wearing jeans or pants in the sanctum sanctorum is wrong. In the footage, a man is seen entering the restricted area wearing jeans. Why was I not allowed? Only because I am a disabled woman? I am a devotee of Lord Shankar," she said.
The 29-year-old mountaineer also said she did not intend to seek an apology from anyone over the incident.
"I am very sorry to tell you that I felt greater pain in visiting Mahakal temple (at Ujjain) than scaling the Everest. My disability was mocked at there (at Mahakal)," Sinha had tweeted yesterday.
She had also tagged the Prime Minister's Office and the Madhya Pradesh chief minister's office in her tweet.
Sinha was pushed out of a running train by robbers in April, 2011, while she was resisting a robbery attempt. As a result, one of her legs had to be amputated below the knee. She is the first Indian amputee to climb the Everest.
In a statement, the minister said she had asked the Ujjain divisional commissioner to conduct an inquiry into the incident.
Chitnis said she had also told the officials that such incidents should not take place in the future.
She added that she would talk to the priests at the temple and also visit Lucknow soon to meet Sinha.
The temple houses one of the 12 "jyotirlingas" in the country.