India had looked to be in the box seat after taking a lead of 190 but Pope's epic 196 resuscitated England, who posted 420 all out in their second innings to roar back into the contest.
That left the hosts needing 231 to win the match but Hartley's 7-62 in his debut test wrecked India, who were all out for 202 in an extended final session on the penultimate day.
Ravichandran Ashwin (28) and Srikar Bharat (28) added 57 runs for the eighth wicket to try to take the match into its final day but that was not to be.
"Incredibly proud," England captain Ben Stokes said of their remarkable fightback against India, who have not lost a test series at home since 2012.
"Before the series started, we knew how much of a beast India are at home but the way we responded to being so far behind, it was testament to everything we have said and lived and breathed over the last two years."
The second match of the five-test series, in Visakhapatnam, begins on Friday.
England, having already conceded a chunky lead, had risked defeat inside three days when they slumped to 163-6 on Saturday.
England vice captain Pope went on to conjure up one of the finest knocks by a visiting batter as his team roared back into the contest.
Pope's knock, spread over six-and-a-half hours, was a masterclass in how to tame India's formidable spin attack in conditions tailor-made for their craft.
The 26-year-old looked immune to any pressure as he swept and scooped to disrupt the rhythm of the Indian spinners and their field placements.
He got terrific support from England's lower order.
Rehan Ahmed (28), playing his second test, and debutant Hartley (34) frustrated India with their solid batting.
Pope's innings came to an end when he attempted a reverse-scoop against Jasprit Bumrah (4-41) and lost his stump in the process.
"To come here to India, toughest place as a batter, and put in a winning performance, it's head and shoulders above the rest," Pope said after collecting the player-of-the-match award.
The Englishman continued to hurt the hosts even as a close-in fielder as Hartley ran through the Indian batting line-up.
Yashasvi Jaiswal (15) stepped out to flick him to short leg and, two balls later, Shubman Gill perished at silly point with Pope taking both catches.
Rohit Sharma (39) had looked assured at the other end but Hartley dismissed him lbw to reduce India to 62-3.
Rahul (22) and Axar Patel (17) briefly resisted England, who were in charge after a tumbling Ben Stokes ran out Ravindra Jadeja with a backhanded flick from mid-on.
Hartley fittingly sealed England's win when he had Mohammed Siraj stumped.
"We have played four whole days of cricket, so it is hard to pinpoint one moment where it went wrong for us," a dejected Rohit said.
"Obviously, having taken a lead of 190 we thought we were really in the game.
"I definitely thought 230 was gettable. There wasn't loads in the pitch but we just didn't bat well enough."
