According to information received here, the trio shot a combined 1938 points to finish four points behind toppers Chinese Taipei in the qualifiers, while Russia (1940) finished second.
But their third place finish meant the Indians will not have to face heavyweights Chinese Taipei and fifth-placed Japan till the final round.
"Initially we were tense. But we ended up being in a good draw," Banerjee, who logged 642 points despite shooting with borrowed bow and arrows, said from Ogden.
Banerjee's shooting equipments has been lost in transit to the US with the Archery Association of India making desperate attempts to trace them before the qualifiers.
India will take on Norway in the first round of the 16-team draw, while in the quarters they will meet the winners of Canada and the Netherlands.
One of the four -- Australia, Turkey, Romania and Russia -- will be India's probable semifinal opponents.
The World Cup Stage III provides the archers one last chance to earn quota places for next month's London Olympics as qualifiers are held along side the main event.
The top three earn the Olympic quota berths.
Braving cold and breezy conditions with temperature in the range of 14 degree celsius, Talukdar shot a fine 653, followed by Rai (643) and Banerjee to finish 14th, 34th and 35th positions respectively in the individual ranking rounds of the World Cup.
Their combined score gave India an overall seventh in the World Cup and third in the Olympic qualifiers.
