Set a target of 245 runs to win, India lost their first three wickets for just 22 runs with James Anderson claiming two and Stuart Broad picking up the wicket of KL Rahul.
Earlier England had set India a challenging target of 245 runs on the fourth day of the fourth Test at Southampton on Sunday.
Resuming on their overnight score of 260 for eight, England lost their ninth wicket on the first ball as Mohammed Shami had Stuart Broad caught behind for a golden duck.
For the second time in the match, Shami was on a hat-trick after he had removed Adil Rashid with the last ball of Saturday’s play.
However, Sam Curran survived the hat-trick ball. England’s second innings ended on 271 runs after Curran was run out on 46 runs.
Shami led India's attack with four wickets for 57 runs from 16 overs.
India have scored 200 or more to win in the fourth innings of a Test outside of Asia just three times.