South Korea will face Australia in the last eight while the Uzbeks play hosts and holders Qatar after beating Thailand 2-1.
Second-choice goalkeeper Jo Hyeon-woo was the hero for South Korea in Doha, saving penalties from Sami Al-Najei and Abdulrahman Ghareeb in the shootout.
Jo was only playing because of a serious knee injury to regular stopper Kim Seung-gyu in training earlier in the tournament.
After a nail-biting game ended 1-1, Hwang Hee-chan scored the decisive penalty for Jurgen Klinsmann's side, who are aiming to end a 64-year Asian Cup title drought for South Korea.
Saudi coach Roberto Mancini disappeared down the tunnel before Hwang took the winning penalty.
"I think this win gives the team an enormous amount of energy," said Klinsmann, who faced fierce criticism in South Korea following draws with Jordan and Malaysia in the group phase.
"It gives the team even more spirit. This is a group of players that I really enjoy working with because they have a lot of heart."
His opposite number Mancini said he had not meant "to disrespect anyone" by leaving the pitch before the shootout had concluded.
"I apologise, I thought it was finished," the Italian said.
In front of a crowd of 40,000 at Education City Stadium, Abdullah Radif opened the scoring for the Saudis 33 seconds after coming on as a half-time substitute.
South Korea had several chances to equalise but left it until the death to do so, sub Cho Gue-sung popping up in the 99th minute to rescue his team and force extra time.
Klinsmann's men dominated both periods of extra time but were guilty of over-elaborating and could not put the ball in the net.
Saudi Pro League sides have shelled out to lure big-name players while Saudi Arabia is set to host the 2034 World Cup, all part of a broader move into global sport.
But they are out of the Asian Cup in the last 16 and their wait for a fourth continental crown -- and first since 1996 -- goes on.
'Why not?'
Srecko Katanec said that his Uzbekistan side can "surprise" Qatar in the quarter-finals despite being depleted by injury and illness.
The Uzbeks took the lead against Thailand in the first half with a cool Azizbek Turgunboev finish through the legs of the goalkeeper.
Thailand were the lowest-ranked team left at the tournament but they hit back just before the hour with an even better goal, from distance, by substitute Supachok Sarachat.
Uzbekistan went ahead again midway through the second half when highly rated 20-year-old CSKA Moscow attacker Abbosbek Fayzullaev struck from outside the box.
This time Thailand had no reply, despite some late pressure.
Uzbek coach Katanec felt his side should have scored two more goals in the first half but was delighted with the result, given they were missing several first-choice players.
"Imagine what kind of work I have to do," said the Slovenian, adding that he is missing five players who would normally start.
None will be available to face Qatar on Saturday, he said.
"We also have some virus, many players are ill, but we performed well and our players did their job," he said.
Despite being short of his best side Katanec struck a defiant tone, saying: "We can surprise Qatar, why not?"
Pre-tournament favourites Japan enter the last-16 fray on Wednesday when they play Bahrain, while Iran face Syria.
