Root was looking set to get a big one but a brilliant piece of fielding from Kohli saw the right-hander getting run-out for 80.
Resuming at 76 for three after lunch, the visitors quickly found themselves on the back foot as all rounder Ben Stokes picked up two wickets to leave India reeling at 100-5 and trailing England's first innings total by 127 runs.
The England skipper was short of his ground following a marvellous piece of athleticism from Kohli, who collected the ball, turned and threw down the stumps in one motion.
But it is the suggestion that Kohli may have sworn in Root's direction that could land the India skipper in trouble with match referee Jeff Crowe, the former New Zealand captain.
The 29-year-old then struck several stylish attacking shots, including a classic cover-drive for four off Stokes and a lofted six off leg-spinner Adil Rashid.
Chasing England's first inning total of 287, India lost 3 early wickets and at one time, they were reeling at 59/3.
Off-spinner Ravichandran Ashwin took four wickets for 60 runs in 25 overs while paceman Mohammed Shami had two for 64 in 19.
But Kohli just about survived. Virat Kohli and co. will aim to reanalyze their plans and strive to come back stronger. It brought out Kohli who was booed on his walk to the crease.
Kohli celebrated the dismissal with a "mic drop" gesture, pretending to drop an imaginary microphone, just as Root had done a month earlier when he clinched the one-day global series with a second successive hundred. 50-0 quickly became 59-3 as Curran took three wickets in eight balls, dismissing each of India's top three.
England will hope Jos Buttler will be able to bat on Friday after he went to the hospital during the lunch interval for an X-ray on his left middle finger, which he injured while fielding. Pitched outside off, Bairstow gets a thick edge which Shikhar Dhawan holds on to at slips.
Curran is an exceptionally promising all-rounder but he was certainly not guaranteed a place in the XI and may not have featured if England had more faith in the fitness of Mark Wood and Chris Woakes.