Kagawa struck with just eight minutes gone before adding a second on the half hour, with close-season signing Andre Schuerrle netting his first goal since arriving from Wolfsburg on the stroke of halftime.
Croatia midfielder Alen Halilovic, who joined from Barcelona last month, grabbed the winning goal for Hamburg as they laboured to a 1-0 win at third-tier Zwickau.
Werder Bremen and newly-promoted RB Leipzig were the only two Bundesliga casualties from the opening round, with defending champions Bayern Munich cruising to a 5-0 demolition of Carl Zeiss Jena on Friday courtesy of a Robert Lewandowski hat-trick.
The draw for the next round of the competition will be made on Friday following the opening game of the Bundesliga season between Bayern and Werder at the Allianz Arena.
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