The pair put on 198 to take England to 496 on the fourth day in reply to India's first innings 457.
At tea, India were 57 for one in their second innings, a lead of 18 runs, with Murali Vijay 19 not out and Cheteshwar Pujara eight not out.
Root made 154 not out - his third score in excess of 150 in 18 matches at this level.
Meanwhile, Anderson's personal best 81 was the third highest score by any Test match No 11 after Ashton Agar's 98 against England at Trent Bridge last year and Tino Best's 95 for the West Indies, also against England, at Edgbaston in 2012.
The England duo's stand surpassed the previous tenth-wicket Test record of 163 - shared by Agar and Phil Hughes at Trent Bridge, last year.
It was the second impressive tenth-wicket stand of the match after Bhuvneshwar Kumar (58) and Mohammed Shami (51 not out) had put on 111 for India in a first innings total of 457.
And it meant this match was also the first in 137 years of Test cricket where both sides had seen their last-wicket duo share hundred partnerships and both number elevens - Shami and Anderson - had made fifties.
As India began their second innings, Vijay, who made 146 in the first innings, should have been out for nought when he edged England spearhead Anderson, only for wicketkeeper Matt Prior to fail to get even a hand on the low chance.
The ball may have dipped a touch on Prior but it was a catch any wicketkeeper at this level ought to have held.
But England did manage a wicket when part-time off-spinner Moeen Ali struck with his fourth ball to have a disbelieving Shikhar Dhawan caught and bowled off a full toss for a run-a-ball 29.
England, after losing six wickets for 68 runs in the face of accurate seam bowling from Kumar and Ishant Sharma on Friday, resumed on 352 for nine.
Root was 78 not out and Anderson 23 not out.
But neither Kumar nor Sharma was able to generate the same reverse swing they'd managed on Friday and that made life far easier for England's tailend pair on a still docile pitch.
Left-handed batsman Anderson was the initial aggressor on Saturday, turning Kumar of his legs for four and cover-driving him for another boundary with shots worthy of a top-order batsman.
Then Root's two boundaries in as many balls, a cover-drive and a a square-drive, off Shami saw the 23-year-old to a hundred in 186 balls with 12 boundaries.
Yorkshire's Root, 50 when Anderson, of arch-rivals Lancashire, came in, joyfully hugged his partner as he celebrated his fourth century in 18 Tests.
The 31-year-old Anderson then went past his previous highest Test score of 34 against South Africa in 2008 when he deliberately uppercut Sharma over the slips for four.
Anderson had so nearly batted England to a draw last time out against Sri Lanka at Headingley before being caught off a rising delivery off the penultimate ball of the match.
It was from such a delivery that Anderson gave a chance on 45 when he fended at Shami only for Vijay to drop the low gully catch.
Anderson was soon celebrating a maiden fifty in his 95th Test, pulling Shami to the boundary to reach the landmark in just 61 balls with 11 fours.
With England nine wickets down, lunch was delayed by 30 minutes.
But India still couldn't take England's last wicket.
However, Anderson was still on 81 when he edged a drive off Kumar and Dhawan at first slip held a fine catch to end a near four-hour innings.
Kumar's return of five for 82 was his maiden five-wicket Test haul.
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