Navneet Kaur's strike handed India a 3-2 lead in the shoot out and when Anna Toman missed the following shot, it secured India's win. The two teams had ended 1-1 after regulation time.
India had surprised top teams in the competition and they finished their campaign in a superb manner.
England girls were also chasing their first bronze medal, having finished fourth at the last edition in 2009 in Boston, where they had lost the bronze play-off to South Korea.
Rani took the first stroke in the shoot-out and scored straight away. None of the England player could score except Emily Defroand, who cancelled India's lead on both occasions.
Navneet had missed her earlier stroke but she beat the rival goalkeeper in her second attempt as India gained upper hand again.
Anna, who had found the equaliser in the regulation for England, could not rescue her side this time in the pressure situation.
Earlier, the match started off with an early penalty corner to England but Grace Balsdon's shot was deflected before reaching India's goalkeeper Chanu Sanarik Ningombam.
The English girls kept pushing for the opening goal but it were India who would get on the scoreboard first. A counter-attack saw Rani powerfully strike home after 13 minutes of play.