Bremen will host RB Leipzig, which it also faces on Sunday (AEST) in the Bundesliga, on 31 April for a place in the final at Berlin's Olympic Stadium on 14 May.
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 said.
"I think everyone in Bremen will sleep well tonight."
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, which won the last of its six DFB-Pokal titles back in 2009, has yet to concede a goal in its run to the semi-finals.