The real estate arm of the German investor bought the property, in Piłsudski Square, from Aberdeen Asset Management for almost €190m.
Deutsche AWM said it was acting on behalf of a German fund for institutional investors.
The acquisition takes its total assets under management in Poland to more than €1bn.
Completed in 2003, the 38,000sqm office building includes ground-floor retail and was designed by Foster and Partners.
Gianluca Muzzi, head of real estate for Europe, said: “Warsaw’s city centre – one of Central and Eastern Europe’s most vibrant and growing business centres, makes it an attractive investment opportunity.”
Muzzi said the firm would continue to focus on quality assets for its clients.
The deal would have followed in the footsteps of the ING Real Estate Investment Management merger with CBRE Investors, which created the world’s biggest property investment manager.
As previously reported, the division will not look to acquire rivals as part of its growth strategy, according to the banking group’s co-chief executive.
Anshu Jain said the creation of Deutsche AWM needed to go hand in hand with “a high level of humility and respect” for the manager’s well-established competitors.