Although the Rio 2016 champions relinquished their title following an emphatic semi-final loss to Holland on Wednesday, they dug deep and secured a third successive Games podium finish.
The London 2012 bronze medallists had to dig deep, though, leading 2-0, then falling behind before winning it in the final quarter with Grace Balsdon's penalty corner clincher.
Ellie Rayer opened the scoring, then Sarah Robertson made it 2-0, only for India to score three times in four minutes during a frantic second quarter, with Gurjit Kaur scoring twice and Vandana Katariya also netting.
Captain Hollie Pearne-Webb equalised in the third quarter, before Britain showed composure to go back in front and close out a fine victory that sparked scenes of wild celebration.
After a scoreless first quarter, Britain needed less than a minute of the second quarter to deservedly go ahead.
Rayer left the Indian defence floundering with a surging run, and she was awarded the goal, despite her cross being heavily deflected past India's goalkeeper.
Ansley then hit the post as Britain continued to impress, and they went 2-0 up following a brilliant turn and shot by Robertson.
Goalkeeper Maddie Hinch made a sharp reflex save following a rare India attack, but the deficit was then reduced after Kaur struck from a penalty corner, before she repeated the feat two minutes later, making it 2-2.
Remarkably, India added a third goal in four minutes through Katayira. It went to a video referral before being allowed to stand, and Britain suddenly had a huge fight on their hands, trailing 3-2 at half-time.
Britain needed to regroup after such a demoralising end to the second quarter, and they responded impressively, establishing a strong platform before Pearne-Webb hit a powerful shot into the corner of India's net.
Two more penalty corners put the reigning Olympic champions under pressure, though, and Hinch made a fine save as a thrilling encounter continued at stamina-sapping pace, tied 3-3 entering the final quarter.
Then Balsdon struck, Britain kept their composure, and India could find no way back as the bronze medal was secured following an epic encounter.