Root, who was 78 overnight, was unbeaten on 143 while left-handed Anderson (81) scored his maiden Test half-century as England took a slender 28-run lead.
They bettered the previous world record of 163-run stand made by Australia's Phil Hughes and Ashton Agar at this very ground during the Ashes series last year.
The fourth morning began with India desperately looking for the last wicket but they allowed Anderson and Root to build a solid stand by their wayward bowling that made things easy for the hosts.
It is also first match in 137 years of Test cricket that number 11 batsmen of both the teams have scored fifties.
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For India, Mohammed Shami had scored an unbeaten 51 while Anderson is batting at 81 for the hosts. Anderson's is the highest Test score by an England number 11 batsman, beating John Snow's effort of 59 not out against West Indies at The Oval in 1966.
Shami had combined with pave colleague Bhuvaneshwar Kumar for a 111-run partnership for the last wicket that helped India reach a first innings total of 457.
As time went by the ploys kept on failing and the lead got slimmer. Root brought up his hundred in the seventh over of the morning after facing 186 balls and hitting 12 fours.
Two overs later, he brought up the 100-run partnership with Anderson as well as the 400-run mark for England.
In the 119th over of the innings, Anderson celebrated his maiden half-century in Test cricket, off 61 balls with 11 fours. He helped England get past the 450-run mark in the 130th over of the innings as also crossing the 150-run mark in his partnership with Root.
Stuart Binny (0-37), their fifth bowler, bowled just ten overs in the entire innings, showing that the Indian team management might have read the pitch wrong and erred in not playing R Ashwin instead.