The Japan international started the Champions League game on the left but moved to a No.10 role in the second half, a position which, he admits, he found more favourable.
"I had an easier time playing as a shadow top [No.10] than I did on the left wing," he told reporters.
"I need to be more involved in the match and I hope I can do so in our upcoming fixtures. I want to play better, sharper and faster."
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The draw leaves the Red Devils top of Group A, a point clear of Bayer Leverkusen, but Kagawa feels that his team might have done even better and secured all three points.
"I think it was a game we could have won," he added. "If you look at the other groups, too, the number of points you earn in games like these can decide whether or not you advance. If we could convert our chances, we could have won."
Kagawa also insisted that little has changed at the club despite Sir Alex Ferguson leaving Old Trafford after almost 27 years in charge and David Moyes taking over at the helm.
"I don't feel like there are any big differences," he argued. "Their instructions change from match to match, after all."