Jo stepped up earlier in the tournament when regular number one Kim Seung-gyu was injured in training, and the back-up proved he was up to the job after a nail-biting last-16 match in Doha ended 1-1.
Jo said "belief" was the key to stopping spot kicks from Sami Al-Najei and Abdulrahman Ghareeb in the shootout, before Hwang Hee-chan converted the decisive penalty for Jurgen Klinsmann's side.
"The coach told us to believe in the team and I think that's what really paid off," said Jo.
"We gave everything on the pitch, and when it came to a shootout I had confidence I could make saves."
South Korea came back from the dead to keep their hopes of winning the Asian Cup for the first time in 64 years alive.
Cho Gue-sung scored a 99th-minute equaliser in normal time to force extra time, after Abdullah Radif had opened the scoring for the Saudis 33 seconds after coming on as a half-time substitute.
Klinsmann's side had been heavily criticised after coming through the group stage in second place behind Bahrain, after draws with Jordan and Malaysia.
Jo said that the players have vowed to "forget about the past".
"We had belief that if we didn't concede a goal we would score, because we have quality attacking players," he said.
Saudi coach Roberto Mancini disappeared down the tunnel before Hwang took the final penalty.
He later apologised, saying "I thought it was finished".'Even more spirit'
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 now out of the Asian Cup after a tense encounter in front of over 40,000 fans at Education City Stadium.
"In football you can lose, you can win," said Mancini, who won the European Championship with his native Italy in 2021.
"We were playing against a very strong team and we played very well. We lost on a penalty shootout. Penalties are a lottery."
The Saudis had the best chance of the first half when they came agonisingly close to finding the target with three headers in quick succession.
Saleh Al-Shehri hit the crossbar and Ali Lajami did the same with the rebound, before Salem Al-Dawsari finally nodded the ball just wide of the post.
The second half had barely kicked off when substitute Radif made his mark.
Dawsari redirected a pass into the path of Radif and the substitute lashed the ball past Jo.
Klinsmann's side threw everything at the Saudis in a frantic end to the 90 minutes and substitute Cho hit the bar with an injury-time header.
The striker got his reward in the 99th minute, nodding home to send the game into extra time.
"I think this win gives the team an enormous amount of energy," said Klinsmann.
"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."
