The 23-year-old claimed a five-wicket haul in the second innings (5-47), with no Indian fast bowler having claimed more wickets on their maiden Test appearance.
Earlier on Friday, India were bowled out for 453 after resuming day three at Eden Gardens on 6-354.
Man of the match Rohit Sharma (177) fell 23 runs short of a double century in his debut Test innings, while Ravichandran Ashwin was bowled by Shane Shillingford (6-167) on 124 to end his own stubborn resistance.
Trailing by 219 runs, West Indies made a promising start and looked in good shape at 1-101, with Chris Gayle (33) the only victim.
However, Shami tore through the middle order as West Indies crumbled, surrendering the last nine wickets for just 67 runs as they were bowled out for 168.
Ashwin also chipped in with three wickets as India took a 1-0 lead in the two-Test series.
It was Kieran Powell's dismissal, caught lbw by Ashwin for 36, that initiated the tourists' downfall, with Dwayne Bravo (37) and Shivnarine Chanderpaul (31 not out) the only others to offer any resistance.
The rest tumbled for single-figure totals in a weak response from West Indies, who will attempt to put things right when the second Test begins in Mumbai on November 14.
It will mark the end of India batsman Sachin Tendulkar's remarkable Test career, with the 40-year-old having announced his retirement from the game in October.
The match will be Tendulkar's 200th in this format as the 'Little Master' aims to finish on a high.