The team of ex-Wallabies coach Dave Rennie, who dropped three games in a row earlier in the season, are now sixth, just two points outside of the top four on a congested table in which the gap between fourth and seventh is just two points.
Former All Blacks midfielder Ngane Laumape scored Kobe's opening try in the capital, while former England back-rower Nathan Hughes was a tryscorer for the Black Rams.
Defeat left the Black Rams, who are coached by the Inverell-born Peter Hewat, hovering dangerously above the bottom of the table, having won just once to occupy one of the three spots for teams who would be condemned to the post-season relegation series.
Saturday action saw ex-Wallaby centre Curtis Rona score his fifth try of the season as Mitsubishi Heavy Industries Sagamihara Dynaboars beat Shizuoka Blue Revs 53-45, with former Northampton and Dynaboars five-eighth James Grayson posting 28 points for the victors, including two tries.
All Blacks No.10 Richie Mo'unga was also a tryscorer as Toshiba Brave Lupus Tokyo beat Yokohama Canon Eagles 27-7 to remain unbeaten after seven rounds, one point behind competition leaders Saitama Panasonic Wild Knights.
The two meet on March 9.
Defending champions Kubota Spears Funabashi Tokyo Bay ran in nine tries to beat Wallaby Quade Cooper's Hanazono Kintetsu Liners 54-19, with Wales fullback Liam Williams and All Blacks hooker Dane Coles among the tryscorers.
Kintetsu remain winless, one point above bottom side Mie Honda Heat.
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