Shaw was finally caught and bowled by leg-spinner Devendra Bishoo for 134 in the last over before tea.
Prithvi Shaw scored at better than a run a ball before lunch on his test debut in Rajkot.
Overall, Shaw was the fifth youngest opener to score a half-century in his maiden Test innings after 17y Hanif Mohammad, Stollmeyer, Tamim Iqbal and Imran Farhat.
Virender Sehwag, who is well known for his cheeky posts on Twitter, showered praises on the youngster and wrote:"It's been the Shaw show".
Shaw reached his hundred off just 99 deliveries in the second session on day one.. Even after losing the wicket of KL Rahul in the very first over, India is a wonderful situation right now, thanks to Prithvi Shaw and Cheteswara Pujara.
The diminutive right-hander becomes India's second youngest century-maker after Sachin Tendulkar, who in 1990 was 17 years and 112 days when he made 119 not out against in England in Manchester during his ninth Test appearance. Gabriel trapped his opening partner Rahul in front with a sharp incoming delivery to give the West Indies an early breakthrough.
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Aiming to maintain their top spot in ICC Test rankings, India will take on West Indies in 1st Test played at Rajkot.
Shaw earned a reputation as a batting prodigy. This first century is all for him.
Meanwhile, Pujara poked at a wide ball and fell for 86 off 130 balls, giving fast bowler Sherman Lewis his first test wicket.
Kohli and his deputy Ajinkya Rahane (41) added 105 for the fourth wicket to continue India's progress before the latter was caught behind to off-spinner Roston Chase. That completes his 24 Test century.
More than half of his runs came through boundaries, 19 in total.
Shaw, who was 75 not out at lunch, reached the three figures in the afternoon session with a typical punch to the cover, the same way he got his first runs in global cricket.