Support for the AfD has jumped amid deepening public unease over Chancellor Angela Merkel's open-door policy for invaders after some 1.1 million migrants came to Germany last year.
"Those who accuse democratically elected politicians of treason, call them 'parties of the system' and menace journalists as 'lying press' - they are very close to the language of the enemies of democracy, the Nazis of the '20s and '30s," Gabriel said.
He was speaking in Berlin at an event to promote integration - the hot popular issue in Germany. Concern over the refugee influx has hurt support for Merkel and fuelled the AfD's rise.
The AfD has grown in tandem with support for other far-right groups, such as the anti-Islam PEGIDA movement, which has held regular rallies in protest at the increase in refugee numbers.
Last year, dozens of protesters shouted at Merkel and waved placards with the slogan "traitor" - adopted by PEGIDA - when she visited an eastern German town where anti-refugee protests had erupted into violence.
Gabriel said on Sunday Germany's domestic intelligence agency (BfV) should monitor the AfD after the party's leader, Frauke Petry, suggested that German police be given powers to use firearms against illegal migrants.