Root was out for 124 after putting on 179 for the fourth wicket with left-hander Moeen after England were 102-3 at lunch.
Moeen was one run short of his fourth Test hundred at the close on Wednesday with Ben Stokes unbeaten on 19.
England captain Alastair Cook won the toss and had no hesitation in batting first on a Rajkot pitch he regarded as a traditional Indian wicket that would assist the slower bowlers as the game progressed.
A nervy Cook lived a charmed life against India's new-ball pace duo of Mohammed Shami and Umesh Yadav when the left-hander was dropped twice in the first two overs behind the wicket.
The opener seemed to have overcome his jittery start and moved to 21 by the end of the first hour's play before he was adjudged out leg before wicket to spinner Ravindra Jadeja.
Cook was unfortunate to be given out off an incorrect decision from the umpire, with replays showing the ball missing leg stump, and his new opening partner Haseeb Hameed offered little assistance by advising against asking for a review.
It was the first time the Decision Review System (DRS) was being used for a bilateral series in India and England erred a second time when they eventually opted to use the technology.
The 19-year-old Hameed, who became England's youngest opener on his debut, wasted one of the two reviews available to his team when he failed to overturn a leg before decision off Ravichandran Ashwin after compiling a composed knock of 31.
Ben Duckett, who opened for England in last month's series in Bangladesh, was pushed down the order to accommodate Hameed and made a brisk start by hitting Ashwin for three boundaries in four balls.
But India's spearhead gained revenge in his next over when Ajinkya Rahane stooped to complete a smart low catch at slip to send Duckett back to the pavilion for 13 and bring the players in for lunch.
The 25-year-old Root hit 11 boundaries and a six in his 180-ball knock and looked largely untroubled until Yadav struck his pad in the penultimate over before tea with the batsman on 92.
India captain Virat Kohli made his first use of DRS but failed to overturn the on-field not out decision after replays showed the ball was clipping the leg stump.
Root, whose ton was his 11th in Tests and third against India, was finally out to Yadav though it needed confirmation from the television umpire after the paceman seemed to have lost control of the ball while celebrating the return catch.
Moeen provided good support and hit nine fours in his unbeaten knock in the first Test of the five-match series.
Original source: Root, Ali put England in control against India