The OFC Champions League winners had already overcome host club Moghreb Tetouan of Morocco via a penalty shootout in a breathless play-off, but they were expected to struggle against the African champions on Saturday.
However, Ramon Tribulietx's side were well-organised throughout and advanced to a semi-final clash with San Lorenzo - champions of South America - thanks to John Irving's second-half goal.
Setif lay siege to the Auckland goal from there but the semi-professional New Zealanders are now one game away from the final - where they could face giants of the global game and European champions Real Madrid.
Having endured a draining encounter in reaching the quarter-finals, Auckland sat deep and allowed Setif the majority of possession in the opening period, attempting to pick the Algerians off on the counter-attack.
Mohamed Benyettou twice went close for Setif, but they struggled to break down Auckland, who ought to have gone in at the break ahead.
Spaniard Angel Berlanga worked an opening and chipped Setif goalkeeper Sofiane Khedairia, but the ball bounced over via the crossbar.
Having soaked up plenty of pressure in the first half, the part-timers took a stunning lead seven minutes after the break.
New Zealand international Tim Payne's corner from the right was poorly executed, but the ball somehow bobbled to defender Irving, who showed great composure to jink away from his defender and hit a low shot beyond an unsighted Khedairia, who was slow to get down to his right.
Setif immediately poured forward in search of an equaliser but Benyettou found Tamati Williams tough to beat in the Auckland goal - the 30-year-old spread himself to preserve the lead, taking a blow to a sensitive area for his troubles.
Benyettou continued to look Setif's best chance of equalising, but the striker saw an acrobatic effort go wide as the game entered its final 20 minutes.
Ryan de Vries curled a shot wide as Auckland looked to put the contest to bed and substitute Sofiane Younes headed wide when unmarked at other end with Setif's last chance.
The final whistle was greeted with jubilant Auckland celebrations as their momentous run in the competition continued.
