The New Zealanders have now set up a semi-final clash with Adelaide after ending City's A-League Men title defence on Saturday.
Just 30 seconds away from advancing thanks to Guillermo May's first-half header, the hosts were forced into extra time when teenager Medin Memeti launched himself at an Aziz Behich free kick and volleyed a dramatic late equaliser to force extra time.
With fatigue stepping in and both sides desperate for a winner, neither could find the back of the net after 30 minutes of extra time, forcing penalties.
Kicking into the Port end of the pitch, both sides were perfect with their first six attempts before Nathaniel Atkinson saw his effort saved by Michael Woud, allowing Dan Hall to fire down the middle and send the Kiwis through to the semis.
"It's never ideal, going to penalties," golden boot-winning striker Sam Cosgrove told Sky Sport.
"I thought we did enough to win the game, it was a kick in the balls conceding so late.
"But the boys, for 90 minutes plus the added time, some bollocks from the boys."
After ending the regular season in third, the Kiwis will now host Adelaide in the first leg of their semi-final next Saturday, before travelling to South Australia for the decisive second encounter.
Cosgrove, however, is under an injury cloud after coming on as a substitute despite having a ruptured ligament in his ankle.
"Ideally, we'd have got the job done in a normal time, and I wouldn't have had to come on," Cosgrove said.
"I've still got two ligaments in there, just missing the one, so I should be fine for next week."
Top-of-the-table Newcastle, meanwhile, will face the winner of Saturday night's Big Blue between Melbourne Victory and Sydney FC.
Auckland were forced to weather a strong first-half push from the visitors, but after repulsing the pressure without conceding a shot on target took the lead with the last touch of the half when May darted to the near post and met Jesse Randall's corner with a glancing header.
It sent the 11,341 in attendance into raptures and City into overdrive as they sought an equaliser.
Socceroo Aziz Behich picked out Memeti at the far post with a magnificent free kick and the 18-year-old volleyed crisply into the net.
But they couldn't carry on the momentum.
"This one's going to be tough to swallow," Behich told Sky Sport.
"But we'll be back again next year. That's what this football club does. We'll be back fighting for titles and finals."
That may be a little difficult, though, as City face a testing time.
Head coach Aurelio Vidmar is off-contract heading into the offseason as are veterans Mat Leckie, Andreas Kuen and Andrew Nabbout.
The loans of Daniel Arzani, Ryan Teague and potential World Cup bolter Marcus Younis have also come to an end.
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