Chancellor Angela Merkel's cabinet approved the draft law based on the recommendations of a parliamentary committee tasked with investigating the bungled murder probe.
Government spokesman Steffen Seibert said the discovery of a "terrorist cell" in November 2011 believed to be responsible for gunning down 10 people, mainly Turkish immigrants, over seven years had created "shock and dismay in Germany".
"We have the duty to do everything we can to ensure that such things never happen again," Justice Minister Heiko Maas said in a statement.
The law, which must still pass parliament, would require criminal justice authorities to do more to take possible "racist, xenophobic or otherwise dehumanising" motives into account in criminal investigations and sentencing.
Public prosecutors would be told to consider such motives "early on" while probing crimes, while the federal prosecutor's office would be given powers to take over a case when it suspects an extremist motive.
The draft law also untangles the lines of authority among stateinstitutions to prevent cases slipping between the cracks.
The parliamentary probe of the so-called National Socialist Underground (NSU), a neo-Nazi trio blamed for the serial murders carried out between 2000 and 2007, found massive failings in the police investigation in a damning 1,000-page report released last August.
It said that a lack of coordination between authorities in various regions and a presumption that the killers must also have been immigrants allowed the group to act with impunity.
The NSU's sole surviving member, Beate Zschäpe, is on trial, accused of lending vital support to the group's two gunmen, Uwe Böhnhardt and Uwe Bundlos, who died in an apparent murder-suicide in late 2011.
Until then, police and the media had dubbed the nationwide series of gangland-style shootings, committed with the same Ceska handgun, the "doner (kebab) murders", suspecting that Turkish criminal groups were to blame.
German police and domestic intelligence services faced withering criticism for ingrained bias by associating terrorism only with the far-left or Islamists, not right-wing extremists.
The case shook Germany's self-image of having learned the lessons of its Nazi past.