The goal was scored in the first-half by Keisuke Honda.
Both teams had won their first games in the pool heading into the match.
Japan played well but poor finishing stopped them from winning by a bigger margin.
Goal scorer Honda played really well hitting the woodwork on three occasions.
The goal came in the 23rd minute when Jalal Hassan then made a superb point-blank save from Kagawa's shot, from the rebound Honda was sandwiched by two players and referee Alireza Faghani awarded the penalty kick.
Honda rolled the penalty into the bottom right-hand corner to give Japan the 1-0 lead.
The exciting Honda came close to a second goal when his 25-yard shot hit the crossbar.
In the 65th minute Japan should have ended the contest when Nagatomo's low centre across the six-yard box picked out Honda, who hit the post with the goal wide open.
Late in the game Kagawa tested Hassan with a stinging shot from the edge of the box that was saved.
Japan leads the group with six points, three clear of Iraq and Jordan. The four-time champions next play Jordan.