The Buccaneers dominated much of the encounter but were made to work hard by a resilient Durban City side, who frustrated the hosts after being reduced to 10 men in the first half.
After an even opening, Pirates created the first meaningful opportunity in the 20th minute when Thalente Mbatha picked up a loose ball in midfield before advancing and unleashing a long-range effort that forced City goalkeeper Frederick Asare into a save.
Pirates continued searching for the opener and missed a golden opportunity when Yanela Mbuthuma failed to keep his effort on target after connecting with a pinpoint delivery from Patrick Maswanganyi from close range.
Durban City threatened in the 29th minute when Jean Lubumba attempted to beat Sipho Chaine, but his effort went wide.
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The hosts eventually broke the deadlock eight minutes later. Tshepang Moremi was brought down inside the box by Brooklyn Poggenpoel, who was shown a red card for the challenge.
Maswanganyi stepped up and calmly converted the resulting penalty to hand Pirates a 1-0 lead in the 37th minute.
The Buccaneers continued to push for a second before the interval, with Oswin Appollis and Camren Dansin both going close, but they went into the break with a slender advantage.
Despite playing with 10 men, Durban City came out of the second half with renewed purpose and found an equaliser just seven minutes after the restart.
A cross into the Pirates penalty area found Lubumba, who made no mistake from close range to silence the home crowd and made it 1-1.
Pirates responded by turning up the pressure, but Durban City defended stubbornly. Moremi came agonisingly close to restoring the Buccaneers’ lead when his set-piece rattled the woodwork.
With neither side able to find a winner in regulation time, the match went into extra-time.
Pirates continued to search for a breakthrough, with Mbatha trying his luck from distance, while Durban City struggled to create clear-cut opportunities.
The visitors’ best late chance fell to Tashreeq Morris after he controlled a pass from Hashim Domingo, but Pirates defender Lebone Seema intervened to deny the striker an opportunity on goal.
Just when the match appeared destined for a penalty shootout, Lungu stepped up in spectacular fashion.
The Pirates attacker found the decisive breakthrough with a superb volley in the final minute of extra-time, sending the home supporters into wild celebrations.
The late winner booked Pirates’ place in the MTN8 semi-finals and kept alive their hopes of lifting the Wafa Wafa trophy for a fifth consecutive time.