She is the second Indian shooter to bag an Olympic quota place after Commonwealth and Asian Games gold medallist Jitu Rai.
Apurvi shot a total of 185.6 in the finals, to finish behind Pejcic Snjezana of Croatia, who won gold, and Ivana Maksimovic of Serbia, who clinched the silver medal.
However, the other Indian shooters in the fray Jitu, Prakash Nanjappa, Rahi Sarnobat, Annu Raj Singh and Anisa Sayyed failed to make an impression.
In the 25-metre women's pistol qualification, Sarnobat was 20th, Annu Raj 43rd and Anisa a lowly 52nd.
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In Apurvi's event, Pejcic eventually finished with a score of 209.1 to Ivana's 207.7 at the end of the gold medal match up. There were three Olympic quota places to be bagged in the event.
"A World Cup medal was something that I have been aiming for a long time, and now that I have it, it really makes e very happy. What adds on to the happiness is winning the second quota place for my country for the Rio Olympics," said an elated Apurvi after winning her medal today.
"This win has helped me feel more confident towards my preparation for the Rio Olympics and I will work towards securing more podium finishes in the competitions I participate in this year," she said.
"I would also like to thank the NRAI for all the support it has given me to achieve this quota place and medal," she added.