Gamba goes through 3-2 on aggregate after the goalless first leg in Jeonju.
Guangzhou Evergande of China awaits it in the semi-finals.
Attacking midfielder Leonardo gave the visitor a shock 13th-minute lead from the spot after Daiki Niwa handled the ball in the penalty area but striker Patric promptly equalised for Gamba.
Shu Kurata put Gamba ahead with a long-range shot in the 76th minute.
Urko Vera headed in a dramatic equaliser two minutes from time for what looked like the tie-winning goal.
But Gamba responded with an injury-time goal from Koki Yanekura.
Guangzhou and Al Hilal of Saudi Arabia had qualified for the semi-finals the previous day.
The fourth team to reach the last four will be decided by Thursday's remaining quarter-final in Dubai between Al Ahli of the United Arab Emirates and Iran's Naft Tehran.
Al Ahli holds a 1-0 lead after the first leg in Iran.
