Sharma stole the thunder with his unbeaten 127 as the hosts, who were precariously placed at 83 for five at one stage, reached a comfortable 354 for six at close on an eventful second day's play.
Ravichandran Ashwin was the other batting hero of the day as he remained unbeaten on 92 as he combined with Sharma to pull India out of trouble. Sharma became the 14th Indian to score a century on debut.
With retiring legend Sachin Tendulkar being given out in a contentious manner in his penultimate match, the in-form Sharma along with Ashwin added 198 runs for the seventh wicket to almost bat the visiting side out of the match on a tricky Eden Gardens strip.
Handed the Test cap by Tendulkar after his Man-of-the Series performance in the recently-concluded ODI series against Australia, Sharma struck 16 fours and one six while facing 228 balls.
The 26-year-old Mumbai right-hander was playing in his maiden Test three years and nine months after playing his first ODI.
Ashwin played the perfect foil as the offspinner made invaluable runs in their match-turning partnership that helped India get past the West Indies first innings score of 234.
On a pitch that has started to crumble, the match is tilted in favour of India, who will be looking to seal the issue without needing to play on the final day.
For the elegant Mumbai batsman, who was assured of a Test debut on the morning of February 6, 2010 against South Africa before a freak collision with Wriddhiman Saha dashed his hopes, it was perseverance that paid off.
Having come after the unlucky dismissal of Tendulkar, Sharma showed amazing resillience.
As wickets started to tumble with Shane Shillingford (4/130) reducing the hosts to 83/5 in the first session, it was left to Sharma and Ashwin to do the recovery act.