On the third day at Eden Gardens today, homeboy Shami grabbed five wickets in the second innings as the visitors were bundled out for 168 in 54.1 overs.
Shami finished with match figures of 9/118, second best by an Indian bowler on debut behind Narendra Hirwani's 16/136.
Rohit Sharma, another debutant who shone with the bat with 177, won the Man-of-the-match award.
The second and final Test of the series begins in Mumbai on November 14. This is will be Sachin Tendulkar's 200th and last.
Here are the pictures from the match.
Mohammed Shami smiles after taking a wicket.
Sachin Tendulkar did not get a chance to bat in second innnings but was called on to bowl few overs by captain MS Dhoni.
Test ton for Ashwin
R Ashwin celebrates his second Test century.
Rohit scores 177
Rohit Sharma raises his bat after crossing 150. He was out for 177. He won the Man-of-the-match award.
Master Blaster Sachin Tendulkar along with former captain Sourav Ganguly releasing tricoloured balloons carrying a poster of him at Eden Gardens.
Ashwin is ecstatic
Ashwin reacts after reaching his hundred at Eden Gardens on the third day.
Gayle fails again
Chris Gayle failed with the bat again and Bhuvneshwar Kumar got his wicket for the second time in the match. Gayle made 33. His first innings score was 18.
Mohammed Shami took five wickets in the second innings to give India victory. He ended the match with 9/118.
Ashwin takes a wicket
R Ashwin celebrates with his team-mates after taking a wicket.
Bravo plays a shot
Darren Bravo plays a shot. He scored 37.
Bravo is disappointed
Bravo is disappointed after he got out to Ashwin for 37.
Vijay and Shami
Murali Vijay and Shami celebrate the dismissal of Denesh Ramdin. Vijay took the catch off Shami.