Kubo took advantage of a hesitation from the Mexican defence to net a powerful shot in the sixth minute. Doan then converted from the penalty spot.
Although Mexico had Yohan Vazquez sent off in the 67th, Roberto Alvarado still gave Mexico brief hope of a comeback by scoring in the 86th from a free kick. Vladimir Lorona had a late header saved.
Japan have a perfect record in Group A after winning both matches at its home Olympics. Mexico are second with three points, ahead of France on goal difference.
South Africa are without a point after losing to France 4-3 in the other group game.