Patel stormed to 67 runs off 54 balls as India overhauled the 103-run target a short while after the tea break on the fourth day.
Indian skipper Virat Kohli remained unbeaten on six runs at the other end. Patel hit a boundary to overhaul the target as India finished at 104/2.
Pacer Chris Woakes and leg-spinner Adil Rashid bagged a wicket each for England.
India now lead the five-match Test series 2-0. The hosts had won the second Test in Visakhapatnam by 246 runs while the first match of the series had ended in a draw at Rajkot.
Earlier, resuming at the overnight score of 78/4, England lost quick wickets to end their second innings at 236 runs.
Haseeb Hameed posted a half-century in difficult conditions to remain unbeaten on 59 and help take England past the 200-run mark.
The visitors were all out for 283 runs in their first innings while India had posted 417.