Rana (66kg) went down to Kazakhstan's Yerbol Konyratov 0-1 1-0 0-1 in a close repechage encounter at the K D Jadhav Wrestling Stadium.
Rana lost the opening period 0-1 after the Kazakh was handed a point on the second foul committed by him.
A visibly upset Rana came back strongly to dominate the second period. He, too, was awarded a point for fouls committed by the Kazakh wrestler.
The local favourite won the toss and was given an advantage. He, however, failed to capitalise and conceded a point to end India's medal hope on the first day of the championships.
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The Greco-Roman style is different from the traditional and popular Indian style, which requires the use of only arms and upper bodies to attack.
"We came very close to winning a medal in this match. Had the Kazakh won the toss, we would have been at an advantage as both wrestlers were tired," said India's coach Hargobind Singh.
In the semi-final, he was handed 0-7 0-4 pounding by Ji-Hun Kim of Korea who then went on to clinch the gold in the 66kg category by beating Iran's M Zeidvand 1-0 1-0 in the finals.
Another bronze medal contender, Sharma, (55kg) was no match for DPR Korea's Won Chul Yun, who defeated the local lad 0-6 0-7 in a one-sided contest that lasted around two minutes.