MPs voted by a broad majority for new legislation proposed by Sebastian Kurz's conservative-far-right coalition government before it was deposed in a corruption scandal in May.
Under the new law, the children, grandchildren and great-grandchildren of those who fled the Nazis can apply for citizenship. Until now only Holocaust survivors themselves could obtain Austrian nationality.
Vienna's Jewish community leader Oskar Deutsch welcomed "a decision that puts Austria in line with its historic responsibility".
Kurz's People's Party (OeVP), which is hoping for another strong mandate in September 29 national elections, made several diplomatic gestures to Israel in the 18 months it governed together with junior coalition partner, the far-right Freedom Party (FPOe).
The new law applies to descendants of those who were Austrian when they fled, or the nationality of another country under the Austro-Hungarian empire that extended from what is now the Czech Republic to beyond Croatia until 1918.
Under a special regime, Vienna, which does not normally recognised double nationality, will grant passports without the need to reside in Austria or give up a previous nationality.
During the debates on the bill in parliament, Austrian officials said second and third-generation relatives of those who fled had inquired about the proposed changes, and particuarly those with British nationality as Brexit looms.
A Viennese Jewish Community employee said the centre did not have any numbers for how many people might be eligible to apply for Austrian citizenship.