The 23-year-old Dipa, who had become the first Indian woman gymnast to have qualified for the Olympics, scored an average of 15.066 points, a mere 0.15 less than the eventual bronze winner Giulia Steingruber (15.216) of Switzerland.
The gold went to American pre-event favourite Simone Biles (15.966), adding to the two gold she has already won in team event and all-round in this Olympics.
Reigning vault world champion Maria Paseka of Russia got the silver with an average of 15.253 points.
Dipa, who had scored 14.850 in the qualifying round, could have got more points had she not landed on her haunches in her risky 'Produnova' vault for which she is now known all over the world.
Coming in as the sixth competitor in the finals, the Tripura girl went for the 'Tsukahara' in her first attempt in which she collected 14.866 points. She went for a difficulty level of 6 and got 8.855 for the execution.
In her 'Produnova', Dipa did everything right except that she did not have a perfect landing as her buttocks just touched the ground and thereby she lost points.
She scored 15.266 from her 'Produnova' with 8.266 points for execution for a difficulty level of 7. The average of the two attempts gave her 15.066, a mere 0.15 difference with bronze winner Steingruber.
Dipa, who won a bronze in the 2014 Commonwealth Games, had some anxious moments as she waited for her score to come on the scoreboard.
Her coach Biswesar Nandi hugged her after the 'Produnova' vault and Dipa herself gave a thumbs up to the camera but it seemed that it dawned on both of them that it was not a perfect landing in the most risky vault which very few in the world attempted.
Dipa was the second competitor to perform the 'Produnova' in the finals, the other being 2008 Beijing vault silver winner Oksana Chusovitina of Uzbekistan, who finished seventh in the eight-woman finals with an average point of 14.833.
Dipa, however, will hold her head high as she has scripted history by becoming the first India to qualify for the vault finals in her first Olympics.
Apart from being the first Indian woman, she was also the first Indian gymnast to have qualified for the Olympics after 52 long years.
Since independence, 11 Indian male gymnasts have taken part in the Olympics (two in 1952, three in 1956 and six in 1964), but this was the first for an Indian woman at the Olympics.