44-kg grappler, Sheetal, leading 2-0 at the end of first period, lost the repechage round bout 2-3 at Arena Armeetz in Sofia, Bulgaria.
Rina kept the Indian grappler at bay after securing lead early in the second period and claimed bronze.
Earlier, Sheetal started off well as she romped over Angelika Ewa Dytrych (POL) and followed it up with a victory over Mariya Georgiyeva (UKR) in the quarterfinals.
However, the Asian Junior bronze medallist could not breach the defenses of Erin Simone Golston of the US in the semifinals.
Ritu Malik (59kg) too lost in the repechages to Belarus' Krystsina Tsan to end her hopes of getting a shot at the bronze in the event. She won her opening match against Cemile Unudan of Turkey.
Anuradha Tomar (67kg) lost in the quarterfinals to Italy's Dalma Caneva as India failed to open their account on the third of the championships.
Meanwhile, Mamta Rani (48kg), Pooja Dhanda (59kg), Sarita (63kg) and Manisha Mahadev Divekar (72kg) will be seen in action tomorrow.
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