The 29-year-old shot a total of 216.7 in the eight-man final to finish third on the podium.
The Asian Games and World Championships silver-medallist was languishing at the seventh position after the end of the first competition series that included an 8.8.
But he clawed his way back with two 10.6 and a 10 in the second competition series. At the end of that series, Rai was placed sixth with a total of 98.7.
The Army man roared back to contention with a couple of 10.6 and ensured a bronze with a 9.9, finishing ahead of China's Zhanyi Xu (197.9).
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With a lead of 0.1 point over Vietnam's Xuang Vinh Hoang, Rai was in the hunt for a silver but an 8.6 meant he had to be content with the bronze.
The gold went to Japan's Tomoyuki Matsuda with a world record total of 240.1 points, while Vietnam's Xuan Vinh Hoang claimed silver with 236.6 points.