Playing his first test in eight years, Parthiv Patel smashed an unbeaten 67 off 54 balls and the hosts chased down their 103-run victory target for the loss of Murali Vijay and Cheteshwar Pujara.
3rd Test: Jonny Bairstow's heroics rally England against India
After conceding a 134-run first-innings lead, England had their backs firmly against the wall and Monday's top-order collapse left them with an impossible task to save the game.
Root's composed 78 helped erased the deficit and an unbeaten 59 by Hameed, batting down the order after breaking his finger, ensured India had a three-figure target to chase after bowling out England for 236 in the afternoon session.
When the visitors resumed on 78-4, nightwatchman Gareth Batty failed to trouble the scorers, trapped lbw by Ravindra Jadeja in the second over of the day.
India's Jadeja has no regrets about missed ton
Jos Buttler, arriving at the crease with his team reeling on 78-5, clobbered Jadeja over his head for six and smacked a boundary to bring up England's 100.
But he fell for a run-a-ball 18 trying to hit Jayant Yadav over mid-on, Jadeja taking a stooping catch at mid-wicket.
Root survived two leg-before appeals in one Jadeja over but otherwise looked rock solid as he brought up his 25th Test fifty with an elegant boundary, raising his bat in muted celebration.
His brave 179-ball battle ended in a rather soft dismissal when he pushed at a Jadeja delivery and Ajinkya Rahane took a sharp one-handed catch at slip.
Hameed refused to give up, however, defying the Indian bowlers with solid defence alongside Chris Woakes who made a fluent 30.
Paceman Mohammed Shami then struck twice in an over with the new ball to all but end England's resistance, bouncing out Woakes and inducing Adil Rashid into an awkward pull that was caught on the boundary.
Hameed was left stranded when James Anderson was run out for five. The 19-year-old remained unbeaten after nearly three hours at the crease to enhance his burgeoning reputation.
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