Ramon Tribulietx's men prevailed after regulation and injury time finished scoreless, dumping out Moghreb in Rabat on Wednesday.
Mehdi Khallati missed the decisive spot-kick for Moghreb, after team-mate and goalkeeper Mohamed El Yousfi had brought his side back in it by saving Auckland's fourth penalty kick, taken by Mario Bilen.
El Yousfi's save on Bilen helped square the shoot-out at 3-3 after four kicks each, as Ahmed Jahouh had the first penalty of the shootout saved by Auckland custodian Tamati Williams.
Sanni Issa put Auckland back in front at 4-3, before Khallati's kick came back off an upright to send the New Zealand club through to a last eight clash with ES Setif on Saturday.
Earlier, Auckland started on the front foot and forced Moghreb keeper El Yousfi into a good early save low down to his left.
But the Kiwi club failed to take advantage of their early dominance, and they were nearly punished on the 25-minute mark.
Moghreb striker Mouhssine Iajour raced onto a throughball, but could only shoot straight at Williams after skipping away from a desperate, last-ditch Auckland challenge.
Emiliano Tade should have given Auckland the lead before the break, but fired wide after a slick move from the OFC Champions League winners.
Moghreb boss Abdelaziz El Amri introduced Zaid Krouch at the interval in an attempt to increase his team's attacking potency, but it was Auckland who - once again - began the half strongest.
As in the first period, though, there was a distinct lack of quality in the final third, but Ryan De Vries could have done better when shooting into the side-netting from inside the area with 15 minutes remaining.
The locals made much of the running in extra time, with Salman Ouald El Haj fouled on the edge of the area by Angel Berlanga - the Spaniard earning the first and only yellow card of the contest.
Moghreb could not create any danger from the free-kick, Ahmed Jahouh's effort blocked by the wall, before Auckland had a fine chance in the second half of the additional 30 minutes to steal victory.
De Vries' cross to the back post found Berlanga, who powered a header wide of the target.
It did not matter, though, as Auckland advanced following a thrilling shootout.
