Chinese Yan Wang defended her title with 14.988 points followed by Japanese Sae Miyakawa with 14.812 points.
Karmakar, who won a bronze in the 2014 Commonwealth Games in Glasgow, stood third at the podium with 14.725 points.
"#DipaKarmakar won #bronze n opened #India's medal tally at d #AsianGymnastics Championship, #Japan #SAI Way 2 go!!" Sports Authority of India (SAI) wrote on Twitter.
Image: Dipa Karmakar
Photograph: Robert Cianflone/Getty Images