Sanni Issa was the hero as he secured victory for the part-timers, who became the first club from Oceania to win a medal in the global tournament.
Ryan De Vries put Auckland ahead just before the break, only for Joao Rojas to restore parity in the second half, with the Mexican side improving immeasurably after half-time.
But Cruz Azul were unable to make the most of their superiority and Auckland held on to force a shootout, with Jacob Spoonley saving Ismael Veladez's kick to set Issa up for the winner.
And the Nigerian obliged, beating Jose Jesus Corona from 12 yards to clinch a famous win and secure third place in the competition.
Both sides struggled to take control of the contest during the early proceedings, with neither looking particularly threatening going forward.
But Auckland did find themselves presented with a fine chance in the 27th minute, as Emiliano Tade skipped around Gerardo Flores in the Cruz Azul area, before firing at Corona from a tight angle.
Tade went close for a second time five minutes before the break, but his tame effort from the edge of the area went narrowly wide of the left-hand post.
And the Argentinian was at the centre of things again in first-half stoppage time as the part-timers took the lead, firing an excellent lofted pass over the Mexican defence for De Vries, who ran through on goal and coolly slotted past Corona.
Cruz Azul began the second half with renewed focus and almost drew level in the 50th minute, as Rojas nodded just wide of the target, before Mauro Formica saw a headed effort of his own disallowed for offside.
Luis Fernando Tena's men were not to be denied in the 57th minute, though, as Rojas pounced on hesitancy in the Auckland defence and slammed into the top-left corner from close range.
The Liga MX side continued to force the issue and saw chances through Formica and Christian Gimenez go begging, sending the game to penalties.
And, after both Formica and Veladez failed to convert their spot-kicks, Issa stepped up and slotted past Corona, sparking scenes of jubilation among the Auckland staff.
