With 15 of their senior internationals touring with the British and Irish Lions in Australia, the young Wales side were outscored two tries to one but they clawed their way back as Asian champions Japan paid the price for missing three kicks.
The hosts, who set the early pace, scored two tries in the match through Michael Broadhurst and Yoshikazu Fujita.
Full-back Ayumu Goromaru converted one and kicked two penalties for Japan but was off the target on three occasions, which proved costly in the end.
The hosts went into the break leading 11-6 but Biggar's four pinpoint penalty kicks kept the Six-Nations champions in the contest. Biggar also converted after Liam Williams' pass set Robinson free for their only try.
Wing Fujita took Japan to sniffing distance of a victory with the team's second try, converted by Goromaru, but a late penalty from replacement Rhys Patchell calmed Wales' nerves and sealed a narrow win for the tourists.
Japan, who have lost all their eight matches to Wales, will face the same opponents on next Saturday in Tokyo.
