The 22-year-old Reddy, who hails from Hyderabad, scored 13.649 average points to win the bronze at the Hisense Arena here.
Tjasa Kysslef of Slovania won the gold with a score of 13.800 while Emily Whitehead of Australia took the silver with 13.699.
Another Indian in the final round, Pranati Nayak finished sixth with a score of 13.416. The sport of gymnastics has a series of World Cup events in a year and they are second in importance after the World Championships.
Dipa Karmakar, who finished fourth in the 2016 Rio Olympics in women's vault event, has won a bronze each in the Asian Championships and 2014 Commonwealth Games but not at the level of a World Cup.
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This was Aruna's first international medal through she has taken part at the 2013 World Artistic Gymnastics Championships, 2014 Commonwealth Games and 2014 Asian Games and the 2017 Asian Championships.
Aruna will also contest in the final round of the floor event tomorrow. In other events, Rakesh Patra finished fourth in the men's rings event. He is still in the fray for the final round of the parallel bars tomorrow.
The 2010 Delhi Commonwealth Games and 2-14 Asian Games bronze medallist Ashish Kumar will also feature in the men's vault event final round tomorrow after he finished sixth in the qualifying round.