Ranked 16th in the world, Conway headed into the -70kg category as an outsider for a podium finish but made a strong start as she disposed of Russia's Ekaterina Denisenkova by ippon inside a minute.
And this continued all the way into her bronze medal contest against Irina Gazieva of Russia, where a throw for waza-ari sealed her the bronze medal with just four seconds remaining.
Conway said: "It's good to be back fighting at this level again and my performance was OK, especially as I've not competed on the tour since Russia in June.
"There's a few things to change before Japan but I feel strong and ready for the challenge of the next two weeks."
Meanwhile, Natalie Powell equalled her career-best result as she picked up -78kg silver.
Powell went up against Audrey Tcheumeo in her final, the same judoka compatriot Gemma Gibbons beat in her London 2012 semi-final en route to claiming silver.
And the Frenchman proved too strong as Powell was thrown for ippon inside a minute and had to settle for silver.
"I'm pleased with my performance, particularly as I beat the Russian that beat me earlier in the year," Powell said.
"It was also good to get onto the mat with Tcheumeo as I have not fought her recently and she is well above me in the rankings.
"Overall, this is a good start to the Asian tour."
Two other GB athletes narrowly missed out on medals as they finished fifth, adding to fifth-place finishes from Kimberley Renicks and sevenths from Nathon Burns, Patrick Dawson and Connie Ramsay on day one.
Tom Reed beat this year's Paris Grand Slam winner Yakho Imamov on his way to the -81kg semi-final but lost to German Sven Maresch, while Sarah Adlington (+78kg) also took fifth in the +78kg event.
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