The duo from US, who had won the mixed team gold in Shanghai World Cup, took a 39-34 lead in the first end and went on to record a close 150-146 win over the fourth seeded Indian pair yesterday.
India will now hope for a bronze medal as Das-Deepika will fight it out against the Mexican team of Aida Roman and Juan Rene Serrano, the bronze medallists in Antalya, in the playoff scheduled later tonight.
Rai overcame a 1-3 deficit to stun no 17 seed Thomas Faucheron of France 6-4 even as his teammate Jayanta Talukdar was stopped by Brazil's Marcos Bortoloto.
Rai, who earlier beat Andres Aguilar Gimpel of China 7-1, will face no 16 seed Thomas Aubert of France in the last-32.
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Fifth seed Atanu Das got a bye into the last 32 and will take on Norwegian Baard Nesteng.
In the women's recurve, 2007 World Cup Final winner Dola Banerjee continued her poor run as she was shown the door by no 33 Marina Canetta Gobbi of Brazil.
Deepika and Laishram Bombayla Devi have been seeded directly into the round of 32.
Deepika Kumari will face Regina Maria Romero of Guatemala, while Bombayla will take on Cuban Maydenia Sarduy.
Meanwhile in the compound division, India qualified sixth in both men's and women's sections.