Jean Patric scored the only goal of the game to earn Takayuki Yoshida's side their third win in four games and ensure Kobe bounced back from losing 4-3 to Gangwon FC in the previous round of matches.
The result takes Kobe back into first place in the 12-team standings as the league phase hit the midway point.
The first eight finishers in both West and East Asia will advance to the knockout rounds in March, with the quarter-finals, semi-finals and final to be played in Saudi Arabia in April.
Kobe had dominated throughout at Misaki Park Stadium with Yuya Osako clipping the top of the crossbar with a shot on the turn in the 35th minute.
But the Japanese outfit had to wait until the 58th minute to go in front when Patric finished off an attacking move with a left-foot effort that bobbled beyond goalkeeper Moon Jung-In.
Meanwhile, Johor Darul Ta'zim climbed up to second with a 3-1 victory over Shanghai Shenhua as the Malaysians registered their second win in a row in the competition.
Xisco Munoz's side were frustrated by Shenhua goalkeeper Bao Yaxiong throughout the opening period, but three minutes after the restart, the hosts took the lead when Jonathan Silva buried Arif Aiman's pass across the face of goal.
Oscar Arribas doubled the lead in the 63rd minute with a fine strike into the bottom corner, having controlled the ball on his chest.
A penalty by Saulo Minero with 19 minutes left gave Shenhua hope of salvaging a point, but Silva thumped his second into the top corner with three minutes remaining to put the result beyond doubt.