England, 2-1 up in the five-match series, were 385 for seven at stumps on the second day -- a first-innings lead of 237 runs after India had been skittled out for just 148 on Friday.
Root was 92 not out and Chris Jordan unbeaten on 19, with the pair having added an unbroken 67 in just 62 balls for the eighth wicket.
Saturday saw Root join two all-time great batsmen in becoming only the third England player to score a half century in every Test of a five-match series after Wally Hammond against South Africa in 1938/39 and Peter May, also against South Africa, in 1955.
John Edrich scored fifties in all six-matches of England's Ashes-winning series against Australia in 1970/71.
India took four wickets in quick succession before tea with off-spinner Ravichandran Ashwin claiming two for three runs in eight balls, as England captain Alastair Cook (79), Gary Ballance (64), Ian Bell (seven) and Moeen Ali (14) were dismissed.
But at the interval England were still 246 for five, 98 runs ahead, with Root 11 not out and Jos Buttler unbeaten on 12 on a day where overhead conditions largely favoured the batsmen and much of the early moisture in the pitch had evaporated.
The pair had put on 80 before wicketkeeper Buttler, bidding to score fifties in his first three Test innings, fell for 45 when he chipped fast bowler Ishant Sharma straight to Ashwin at short mid-wicket.
Root then hooked a short ball from swing specialist Bhuvneshwar Kumar for six before completing a 93-ball fifty.
That followed the 23-year-old Yorkshireman's scores of 154 in the drawn first Test at Trent Bridge, 66 in India's 95-run second Test win at Lord's, 56 in the hosts' 266-run victory at Southampton and 77 in their even more emphatic innings and 54-run success at Old Trafford.
England resumed Saturday on 62 without loss after Mahendra Singh Dhoni, who made 82, had stood firm during India's latest batting collapse of the series.
Cook was 24 not out and Sam Robson 33 not out.
However, Robson had added just four runs to his score when, in the second over of the day, the Australia-born opener had his off stump knocked over by fast bowler Varun Aaron after playing down the wrong line.
But England's all left-handed second-wicket pair of Cook and Ballance were barely troubled as they took the hosts to 148 for one at lunch.
The first ball after the interval saw England go into the lead, Cook leg-glancing Kumar for four.
Ballance then completed a 77-ball fifty when he deliberately uppercut Kumar over the slips for his 11th four of the innings.
That meant the Zimbabwe-born batsman had now scored three hundreds and three fifties in his first eight Tests.
Cook, so fluent before lunch, was dropped in the slips on 65 and 70 by Murali Vijay and Ajinkya Rahane respectively.
But he was unable to score what would have been his first century in 31 innings since the last of his England record 26 Test hundreds, against New Zealand at Headingley in May last year.
Cook's luck ran out when he was drawn into a drive outside off stump by Aaron and edged low to Vijay at first slip.
It was the toughest of the three chances the opening batsman had offered, with England now 191 for two after a second-wicket stand of 125.
Cook's exit sparked a mini collapse, with Ballance prodding Ashwin tamely to Cheteshwar Pujara at silly point.
Sharma, the star of India's 95-run win in the second Test at Lord's but missing at Southampton and Manchester with a leg injury, then produced an excellent full-length delivery that Bell edged to Dhoni.
Ashwin struck again when Ali, trying to withdraw the bat, played on.