Local councillors representing Chancellor Angela Merkel's Christian Democrats (CDU), its governing partner the centre-left Social Democrats (SPD), and the liberal Free Democrats (FDP) all voted for Mr Jagsch.
"The NPD candidate filled the vacuum," said Markus Brando, the SPD leader in Altenstadt, which encompasses Waldsiedlung.
But senior party members have condemned the move, with CDU leader Annegret Kramp-Karrenbauer calling for the election to be cancelled. Its secretary-general, Paul Ziemiak, said "the election of a member of a party which pursues anti-constitutional goals is a disgrace".
SPD Secretary-General Lars Klingbeil tweeted his outrage too: "The SPD has a very clear position: We don't co-operate with Nazis! Never! That goes for the federal level, regions and districts. The decision in #Altenstadt is incomprehensible and unjustifiable. It must be reversed immediately."