Kohli became the first captain to have registered 1,000 or more runs in three consecutive calendar years and he also became the first Indian batsman to have scores 1,000 or more Test runs in three consecutive calendar years.
Fast bowler Mohammed Shami removed both Windies openers, bowling stand-in captain Kraigg Brathwaite and trapping Kieran Powell lbw. Only Sachin Tendulkar and Parthiv Patel were younger than Shaw when they got their maiden Test half-centuries, but not on debut.
18-year-old Prithvi Shaw made best use of the flat track to score a century on Test debut.
He reached his half-century off 56 deliveries and was unbeaten on 75 off 74 deliveries at lunch as India reached 133 for one, with Pujara on 56. Although, he was out for 134 runs in the end before the Tea break.
The Indian skipper also surpassed Australia's Steve Smith to go second in the list of centurions among active players, behind South Africa's Hashim Amla on 28.
It has been a good last 30 days for the southpaw, who scored a gritty 86 in the final Test in England before making an ODI comeback with the Asia Cup in Dubai.
Kohli, who started on his overnight score of 72, got to his hundred in his 123rd innings with a boundary off Bishoo. His innings was laced with 15 boundaries. Subsequently, Shaw offered an easy return catch to leggie Devendra Bishoo trying to play against the turn.
Any hopes West Indies had of finding further immediate breakthroughs were then dashed as Shaw and Pujara combined in a 206-run second wicket stand to consolidate the innings.