Alastair Cook made 79, Gary Ballance 64 and Jos Buttler 45 as England turned the screw after their four-pronged seam attack dismissed the touring side for 148 on the first day.
Root ended on 92 not out with Chris Jordan on 19 and England will look to add quick runs in the morning before attempting to bowl out India again to clinch the series which they lead 2-1.
“It’s great that we have batted all day and given ourselves a good lead,” Ballance told reporters.
“There’s a bit of seam movement and if you bowl in good areas you can create chances. Hopefully, it will still do that tomorrow.
“Rooty tried to put the bowlers under pressure and he did that very well.”
Resuming on 62 for no wicket, England opener Sam Robson fell for 37 in the second over of the day when he misjudged a full straight delivery from Arun Aaron which crashed into his off stump to end a first-wicket stand of 66.
Ballance and Cook played comfortably and accumulated runs steadily with crisp strokes off their legs and the odd well-timed drive.
Cook reached his third fifty in four innings with a square cut for four off Ravichandran Ashwin, confirming his return to form with the bat after a wretched run.
England took lunch on 148 for one but Cook was dropped twice after the interval before pushing at a wide ball from Aaron and being well caught low down by Murali Vijay at first slip to end a second-wicket partnership of 125.
Ballance reached his third fifty in eight tests, to go with three centuries, before prodding a turning delivery from spinner Ravi Ashwin straight to Cheteshwar Pujara at silly mid off.
“I started the season feeling good and I’ve brought that to the test matches,” Ballance said.
“I felt like I was going nicely but then probably played at a ball that I shouldn’t have.”
Ian Bell, on seven, received a snorting ball from Ishant Sharma which he could only edge to wicketkeeper MS Dhoni, the third wicket to fall for the addition of only 13 runs.
Moeen Ali batted fluently, clipping two sweetly-struck fours off his legs, and was unfortunate to be dismissed for 14 when he chopped a ball from Ashwin back on to his stumps.
Root and Buttler took England to tea on 246 for five and they extended a jaunty sixth-wicket partnership to 80 with good running between the wickets and some clean hitting.
Buttler looked on course to make a third fifty in his first three tests when he clipped Sharma straight to Ashwin at short mid-wicket.
Root pulled Bhuvneshwar Kumar over long leg for the first six of the innings before Chris Woakes was caught by Dhoni down the leg side off the same bowler for a duck.
Yorkshireman Root reached his half-century off 93 balls, including a six and two fours, to become the fourth England batsman to pass fifty in every test of a five-match series.
Walter Hammond achieved the feat against South Africa in 1938-39, Peter May did likewise against the South Africans in 1955 and John Edrich made at least a half-century in each of six tests on the 1970-71 Ashes tour of Australia.
Root and Jordan added 67 for the eighth wicket off 62 balls, the former hitting nine fours to move to the verge of his fifth test century. (Reuters)