The Chinese Super League side were joined in the quarter-finals by Japan's Urawa Red Diamonds, who won the title in 2007 and 2017, after they beat South Korea's Ulsan Hyundai 3-0 to secure a 4-2 aggregate win.
Trailing 2-1 from the first leg in Japan, Urawa striker Shinzo Koroki scored a goal in each half against the K League side before Brazilian midfielder Ewerton added a late third.
"I'm very satisfied that we have made it to the next round, so we can relax a little bit," said Urawa coach Tsuyoshi Otsuki who took over last month from Oswaldo Oliveira.
"It was a very difficult game for us but last week we worked very hard and now that hard work has paid off." Shanghai came from behind in South Korea to take the game into extra-time and penalties after World Cup striker Kim Shin-wook put Jeonbuk in front after 27 minutes.
Kim's goal gave Jeonbuk a 2-1 aggregate lead following the 1-1 draw in Shanghai last week, but with former Porto striker Hulk in inspired form SIPG were never likely to go without scoring.
The ex-Brazil international equalised 10 minutes from time and he also hit the woodwork on three occasions. With the scores level at 2-2 on aggregate the game went into extra-time and after a goalless additional period the outcome was decided by a penalty shootout.
Shanghai goalkeeper Yan Junling saved Lee Dong-gook's penalty and, with the visitors perfect from the spot, Oscar netted to secure a 5-3 shootout win for Vitor Pereira's side.
"We saw two strong teams tonight and it was a very difficult game, but I think we showed today that we have a strong spirit," said Pereira.
"We played and we fought and in the end we deserved the result. I'm very happy and very proud of my players." Shanghai joined fellow Chinese Super League side Guangzhou Evergrande in the last eight while defending champions Kashima Antlers are also in the eastern half of the draw alongside compatriots Urawa.
The games in the last 16 phase featuring teams from the west of the continent will be played in August, with the draw for the quarter-finals to be made in Kuala Lumpur on July 2.
(Editing by Ed Osmond)
