This was also the 2nd Test century for Kohli against West Indies and both came as captain - joint 2nd most centuries against West Indies as captain along with Dilip Vengsarkar.
Electing to bat, India were 364 for 4 at stumps.
Indian skipper Virat Kohli on Friday hit his 24th Test Century and surpassed former Australian skipper Steve Smith (23) to record the most number of Test hundreds by an active player.
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.
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Earlier, India got off to a nightmare start, losing KL Rahul for a duck in the very first over of the match.
Shaw just needed 99 balls to get the landmark as he toyed with the weakened West Indies bowling, which missed captain Jason Holder and pacer Kemar Roach.
Indian batsmen Virat Kohli and Rishab Pant return to the pavilion after the end of the test cricket match played between India and West Indies, in Rajkot. Unfortunately, Shaw fell at the stroke of tea, in what was a soft dismissal as he chipped one straight into the hands of leggie Devendra Bishoo.
The Mumbaikar punched the second ball he faced towards the cover boundary for a three, his first runs in worldwide cricket, easing whatever nerves left inside.
Rahane, meanwhile, had faced 92 balls and struck five fours and looked a safe bet for a half-century before edging a cut at part-time off-spinner Roston Chase to be caught at the wicket, with the close in sight.