After England were reduced to 112/5 in the first session of day one on Friday, Root, who took 274 balls to smash 122, partnered with Ben Foakes (47 off 126 balls) to recover the visitors.
Later, Ollie Robinson, who slammed 58 off 96 balls, too joined Root in anchoring the English innings.
Ravindra Jadeja picked four wickets for India conceding 67 runs while debutant Akash Deep, who replaced Jasprit Bumrah in the match, scalped three for 83.
India are leading the five-match Test series 2-1.