Root reached his 23rd Test century as England moved to 423-8 at stumps on day two, having bowled out India on the first day for only 78.
Kohli fell caught behind to Jimmy Anderson for 7 on the opening day, completing 50 international innings since his last century.
The India skipper’s last hundred came back in 2019 against Bangladesh.
But India have bigger problems, given their 345-run deficit after two days, with England having reached 120-0 at stumps on the first day, before ramming home their advantage.
Dawid Malan scored 70, combining with Root for a 139-run third wicket stand after openers Rory Burns and Haseeb Hameed made 61 and 68 respectively.
India will hope to wrap up the tail promptly on Friday, with Jasprit Bumrah having removed Root bowled for 121 from 165 balls, across more than four hours at the crease.
Mohammed Shami made some late inroads and finished play with 87-3 while Mohammed Siraj claimed 86-2 and remained hopeful of an Indian comeback.
“Sometimes we have a bad day, when we get out early in a Test match and have to field longer,” Siraj said.
“But that shouldnt lower your morale as we are still left with two matches and leading the series 1-0.
“We don’t need to worry too much, we just need to believe in our skills and back ourselves.”