Shaw, who is just over a month shy of 19, reached his hundred off just 99 deliveries during a dominant 206-run second wicket partnership with Cheteshwar Pujara (86) in the afternoon session of day one.
He made 134 off 154 deliveries with the help of 19 boundaries before being caught and bowled by leg-spinner Devendra Bishoo in the last over before tea.
Shaw reached the landmark with two runs off paceman Keemo Paul and pumped his fist in response to the home crowd's standing ovation.
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.
"Lovely to see such an attacking knock in your first innings, @prithvishaw! Continue batting fearlessly," Tendulkar said on Twitter.
Other Indian luminaries lined up to tweet his praise including former Test batsman VVS Laxman, who said Shaw had a "bright future" in the game.
"It's just the beginning, the lad has firepower," tweeted former opener Virender Sehwag.
Shaw is the seventh youngest international batsman to score his first Test century. He is also the third batsman after Shikhar Dhawan (85 balls) and Dwayne Smith (93) to hit his debut ton in under 100 balls.
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