The 23-year-old pulled out of the game in Saitama an hour before kick-off having felt a twinge in his lower back during training the previous day.
"I underwent treatment for as long as possible before the match and [Japan coach Alberto Zaccheroni] waited for me," said Kagawa, who joined United for $21 million in the pre-season.
"I felt a twinge and didn't want to take the risk," added the former Borussia Dortmund midfielder, who watched the Group B game from the stands.
"It's not a serious injury and I don't think it will keep me out for long. I can walk without any problem but if I put strain on it it still hurts."
Kagawa is set to undergo tests on his return to Old Trafford, where he has made an impressive start to his United career, although he is likely to be a doubt for Saturday's home game against Wigan Athletic in the Premier League.
"I waited until the afternoon to see how his condition was but there's no way I can put a player on the pitch who is not at 100 percent," said Italian Zaccheroni.
After the weekend encounter United kick off their Champions League campaign at home to Turkey's Galatasaray on September 19 before a trip to fierce rivals Liverpool in the league on September 23.
