Substitute Mauro Matos came off the bench to net a 93rd-minute winner in Marrakech as the Argentine club edged out their New Zealand counterparts to set up a meeting with the Blancos on Sunday.
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Auckland came into the clash as huge underdogs to move into the final, but Ramon Tribulietx's side battled well and came within moments of earning a half-time draw.
However, Pablo Barrientos volleyed home the opener shortly before the whistle to give the Copa Libertadores champions a narrow advantage.
City responded well to falling behind and looked the better side throughout an otherwise lifeless second-half display, and Auckland drew level on 67 minutes through Angel Berlanga.
Being pegged back acted as a wake-up call for Lorenzo, who spurned a few chances to snatch a winner, with Martin Cauteruccio hitting the post from distance, but Tribulietx's side held on to earn an additional 30 minutes.
Their resistance waned just three minutes into extra time, though, as Matos latched on to Barrientos's scuffed effort to smash home what proved the winner.