The tourists were 284 for four at stumps with Ali (120) and Ben Stokes (five) at the crease after they elected to bat first at the city’s MA Chidambaram Stadium.
Ali built crucial partnerships with Root (88) and Jonny Bairstow (49) to revive the England innings after the visitors lost their openers in the first hour of play.
Ali’s 146-run third-wicket stand with Root remained the highlight of the innings even as a persistent Ravindra Jadeja claimed three wickets for India.
Ali, who recorded his second century of the series, piloted the innings after the fall of Root.
In-form Root, who completed his fourth half-century of the series in the post-lunch session, steadied the ship after the fall of Keaton Jennings (1) and skipper Alastair Cook (10).
Root started off patiently but soon took the attack to the opposition, smashing 10 boundaries before being caught behind off Jadeja.
The Englishman was given not out by the on-field umpire but India successfully reviewed the decision and he was given out caught behind.
Ali showed grit during his controlled knock as he carried on the good work with Bairstow, who fell one run short of his half-century.
The left-right batting combination of Ali-Root and then Ali-Bairstow thwarted the Indian bowling attack, which had looked dangerous in the morning session.
Ishant Sharma, back in action after an illness, struck first for the hosts on a moisture-laden pitch.
The lanky fast bowler dismissed Jennings, who impressed on his debut with a century at Mumbai’s Wankhede Stadium, caught behind for one.
England’s all-time leading run-scorer Cook edged past the 11,000-run mark on the first ball of the match, but he quickly fell to Jadeja.
Left-arm spinner Jadeja claimed the England captain’s scalp for the fifth time in the series, with the left-handed opener caught at slip by his opposite number Virat Kohli.
The Indian players wore black arm bands as a mark of respect for the popular Tamil Nadu state chief minister Jayalalithaa Jayaram, who died last week.