Kagawa was left out by Japan coach Alberto Zaccheroni for the 1-0 win as a precaution after suffering a problem with his lower back.
The midfielder is confident the problem is not serious and does not expect to be on the sidelines for too long.
"I felt a twinge in the back during practice when I was on the ball. I had some treatment but didn't want to risk causing the team trouble by playing," Kagawa told the Kyodo news agency.
"Obviously it's a shame I wasn't able to play... It's not that serious and I don't think it will keep me out for long."
United boss Sir Alex Ferguson will be hoping Kagawa can recover from the problem in time for Saturday's Premier League clash with Wigan and the start of the Champions League next week.