Replying to the tourists' total of 234, India slipped to 5-83 on day two in Kolkata.
However, Sharma belied his lack of five-day experience to turn things around in style, striking 127 not out and sharing an unbroken seventh-wicket partnership of 198 with Ravichandran Ashwin.
India were therefore able to reach stumps boasting a lead of 120 on 354-6, with Ashwin unbeaten on 92.
"I'm glad I could manage the day and get the team to a good total," Sharma told Star Sports.
"A Test hundred is always a special one and getting one on debut was more special.
"I'm thankful, (it's) another important day tomorrow (Friday) so let's hope it's a good one. There was a lot of pressure; we lost five early wickets.
"When I get back to the hotel room and relax, I'll look back on what I've done.
"I was very focused and I wanted to do it. I was glad I could manage something like that."
Off-spinner Shane Shillingford was chiefly responsible for India's early struggles, claiming four of the first five wickets to fall.
Shillingford ended the day with 4-130 from 41 overs and offered an optimistic outlook at the close, despite India's fine fightback.
"The match, I think, is still wide open. It's only the second day of the Test," he said.
"We have to be patient and, when we get the guys out, sort out a plan for how we're going to bat."
On his personal success, Shillingford added: "It's a wonderful achievement for me to get the wickets I got early on. I know that the Indians are very good at playing spin."