Ravichandran Ashwin also made it a memorable day by cracking his second Test century and then chipped in with three wickets as the hosts bundled out the visitors for a paltry 168 in the second innings to clinch the match with two days to spare.
Rohit Sharma, too, made his Test debut special, cracking a magnificent 177 in the first of the two-match series which is also retiring legend Sachin Tendulkar's last outing in international cricket.
Apart from Sharma, the star of the match was undoubtedly Shami who claimed 5/47 in the second innings and returned impressive match figures of 9 for 118 at the Eden Gardens.
Shami's effort is the second best by an Indian on debut after Narendra Hirwani's 16/136 against the same opponents in Chennai way back in 1988.
Earlier, Sharma missed out on a double century on debut by 23 runs while Ashwin slammed his second Test ton in a record 280-run stand for the seventh wicket as India extended their lead to a mammoth 219 runs before being bowled out for 453.
Starting their second innings with a huge deficit, West Indies were bundled out for just 168, with off-spinner Ashwin (3/46) providing Shami the support at the other end.
Sharma slammed an exquisite 177 off 301 balls with the help of 23 fours and a six, while Ashwin made his runs off 210 balls, hitting 11 fours.
Making his debut at his home ground, Bengal pacer Shami, who took 4/71 in the first innings, utilised the conditions to maximum effect as he rocked the West Indies batting line-up with his lethal reverse swinging deliveries that skidded in sharply.
It was India's fifth Test victory in a row after their 4-0 rout of Australia as the victory also snapped the West Indies' six-match winning streak.