Commonwealth champion Powell, 25, will be the first Welsh woman to compete in judo at the Olympics after collecting more ranking points than Gibbons, 29.
She beat China's Zhehui Zhang in a -78kg bronze medal contest.
England's Gibbons, who has been hit by injuries, lost her bronze medal match to Guusje Steenhuis of the Netherlands.
Both Powell and Gibbons will meet the qualifying requirement of being ranked in the top 14 in the world, but each nation can only select one judoka for each weight category for this summer's Games in Brazil.
Powell is set to finish eighth in the world, with Gibbons 12th.
Britain's Natalie Powell won bronze at the Mexico World Masters to qualify for Rio 2016, meaning London 2012 silver medallist Gemma Gibbons misses out.