"We can tolerate many things, but not political extremism, right-wing or religiously motivated extremism," the minister, Ursula von der Leyen, told public broadcaster ZDF.
She was reacting to the strange case of a 28-year-old army lieutenant, named by German media as Franco A., who led what prosecutors called a "double life" pretending to be a Syrian refugee.
He was arrested last Wednesday on suspicion of planning a gun attack which he meant to blame on his alter-ego -- a fictitious Damascus fruit seller.
The scandal widened after news magazine Der Spiegel reported the suspect had expressed far-right views in a 2014 academic paper, but that no disciplinary action was taken against him.
The military intelligence service is currently investigating around 280 cases of suspected far-right sympathisers in the German armed forces, the report said.
Von der Leyen pointed to leadership failures within the Bundeswehr and criticised "a misunderstood esprit de corps" that had led superior officers to "look the other way" in the lieutenant's case.
She and Interior Minister Thomas de Maiziere, in charge of immigration and refugee issues, have vowed to clear up the embarrassing case, which has led one Social Democrat member to label them a "security risk" for Germany.
The lieutenant was first temporarily detained in February, by Austrian police at Vienna airport, after he tried to retrieve a loaded, unregistered handgun he had hidden in a toilet there days earlier.
This sparked an investigation in which a fingerprint check threw up an even bigger surprise: the suspect had in December 2015 created a false identity as a Syrian refugee.
The soldier, who has an Italian father and German mother, had pretended to be a Damascus fruit seller named "David Benjamin" -- ostensibly a Catholic with Jewish roots who had fled the Islamic State militant group.
He had registered himself at a German refugee shelter and even launched a request for political asylum, said the prosecution statement. Incredibly, the request was accepted, even though the soldier speaks no Arabic.
He was allotted a place in the refugee home and from January 2016 onward received 400 euros ($435) a month in state assistance under this false identity.
The Bild daily has now reported that police found a "death list" compiled by the suspect, including left-wing anti-fascist activists.
Police last Wednesday also arrested a second German man, a 24-year-old student and alleged co-conspirator named by media as Mathias F, who was reportedly in possession of bullets, flares and other objects that breach weapons laws.
Germany has taken in more than one million asylum-seekers since 2015, many from Syria, Iraq and Afghanistan, sparking an anti-foreigner backlash and a spate of racist hate crimes.