Defending champions China thought they had gone ahead in the 39th minute but Wang Shuang's effort was ruled out by VAR.
With the teams level at 0-0 after 90 minutes, China finally got the breakthrough four minutes into extra time through Shao Ziqin.
China had a golden opportunity to add to their lead 10 minutes later but Wurigumula's penalty was saved. They got their second thanks to an own goal from Taiwan's Chen Ying-hui four minutes later.
With the win, China booked a spot at next year's Women's World Cup in Brazil, while Taiwan will compete with the other losing quarter-finalists for two more places in play-offs on the Gold Coast next week.
"I'm so happy we won the match today and that I was able to score a goal," Shao said.
South Korea put six past Uzbekistan
South Korea found it much easier going in their quarter-final against Uzbekistan, taking the lead in the ninth minute through Son Hwa-yeon, whose shot from a tight angle took a heavy deflection and looped over goalkeeper Maftuna Jonimqulova.
Ten minutes later, defender Ko Yoo-jin picked up the ball on the edge of a crowded penalty area and scored with a curling effort to double South Korea's lead at Sydney's Stadium Australia.
A deflected shot from Park Soo-jeong gave South Korea their third after half-time, with all-time leading scorer Ji So-yun then netting her 75th international goal and Lee Eun-young and Jang Sel-gi also getting on the scoresheet.
South Korea, who have qualified for a fourth straight Women's World Cup, will meet either two-time Asian Cup champions Japan or the Philippines in the semi-finals.
