The 21-year-old attacking midfielder, plucked from the obscurity of Japan's second division for only 350,000 euros, struck once in each half and could have added more as he helped his new team outclass last season's runners-up.
"We will win and I will score two goals," the smiling Kagawa had said days before the big match at Schalke, one of the Bundesliga's most fierce rivalries.
He has scored three league goals this season and has also been on target in the Europa League.
Kagawa, not part of Japan's World Cup team, has seen his market value soar to several million euros since signing a contract that runs to 2013.
It took him only a few minutes to make a mark on the match, forcing Germany and Schalke keeper Manuel Neuer into a double save.
Kagawa went one better in the 20th minute, picking up the ball down the right, gracefully shaking off three defenders and curling a slightly deflected left-foot shot past the helpless Neuer.
Schalke still failed to contain the quick Japanese in the second half and Kagawa, who scored his first league goal last week against VfL Wolfsburg, slipped past the defence and tapped in a Jakub Blaszczykowski cross at the far post.
"This was a well deserved victory," said Borussia coach Jurgen Klopp. "We played an outstanding game."
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