The tourists had declared their second innings at 260-3 shortly after lunch, leaving the home side to chase 310 runs off a minimum 49 overs on the fifth and last day.
But the Indians were soon fighting for survival after losing wickets in a clutch on a wearing track at the Saurashtra Cricket Association ground, hosting its first Test ever.
The Indians finished on 172-6 with local hero Ravindra Jadeja also remaining not out on 32.
England had India in early trouble at 71-4 but Kohli steadied the ship with a 47-run partnership that he shared with Ravichandran Ashwin (32).
Adil Rashid was the pick of the bowlers with 3-64 while his spin partners Moeen Ali and Zafar Ansari took one wicket apiece.
Ali, who scored a century in the first innings, was named man of the match.
England skipper Alastair Cook praised his bowlers for putting India's batsmen under constant pressure.
"It was really tough five days," said Cook.
"It was a good cricket wicket. We were relentless with the ball, we didn't let India score too quickly. Adil (Rashid) stepped up to another level."
India's second innings began on a shaky note, with opener Gautam Gambhir falling to paceman Chris Woakes for a duck.
Leg-spinner Rashid then got going, getting rid of Cheteshwar Pujara (18) who was trapped leg before wicket and Murali Vijay for 31.
Both Pujara and Vijay had hit centuries in India's first innings total of 488 in reply to England's 537.
Ali shattered the stumps of Ajinkya Rahane (one) and Ansari sent back Ashwin, who had proved to be a thorn in England's side the last time they toured India.
Kohli held his ground amid the mayhem, hitting six fours in his crucial 98-ball knock.
