The former captain brought up his 38th hundred while also surpassing the legendary Ricky Ponting to become the second-highest run-scorer in Test history.
Root was unbeaten on 121 alongside skipper Ben Stokes (36 batting) as England took a lead of 75 runs.
Earlier in the session, Washington Sundar struck twice in quick succession, removing Ollie Pope (71) and Harry Brook (3) to give India brief hope.
Brief scores: India 1st innings: 358 all out in 114.1 overs (Sai Sudharsan 61, Yashasvi Jaiswal 58; Ben Stokes 5/72).
England 1st innings: 433/4 in 102 overs (Joe Root 121 batting, Ben Duckett 94; Washington Sundar 2/30 ). PTI APA DDV