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Bremen will host RB Leipzig, who they also face on Saturday in the Bundesliga, on April 30 for a place in the final at Berlin's Olympic Stadium on May 13.
The quarter-final had been postponed from early March when the Regensburg squad were quarantined after a Covid-19 outbreak.
"It wasn't a top performance from us, but we just wanted to get to the semi-finals," Bremen coach Florian Kohfeldt told Sky.
"I think everyone in Bremen will sleep well tonight."
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Having beaten top-flight Cologne on penalties in the last 16 back in February, Regensburg held off Bremen until early in the second half.
Bremen captain Niklas Moisander clattered the woodwork with a header, but the visitors went ahead in the 52nd minute with an excellent piece of skill by Osako.
A long pass from midfielder Marco Friedl picked out Osako in the area and he chested the ball down and volleyed home.
The 30-year-old had a chance to make it two soon after but was denied by Regensburg goalkeeper Alexander Meyer, while team-mate Kevin Moehwald also went close.
Bremen, who won the last of their six German Cup titles back in 2009, have yet to concede a goal in their run to the semi-finals.
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