Livakovic put in a man-of-the-match performance with 10 saves across the 120 minutes before also stopping Rodrygo's penalty in the shootout.
Croatia needed an equaliser in the 116th-minute to force penalties and afterward Livakovic praised the spirit shown in securing a second consecutive World Cup semifinal. "We are fighters and we played our hearts out," he said.
"We're happy about it but we're playing one match at a time and we'll see where it takes us. We are experienced and we are raised as fighters. We spare no effort."
"It's one of our greatest wins," said coach Zlatko Dalic, whose team will now face Argentina or the Netherlands in the semifinals.
"We demonstrated what it means to never surrender and demonstrated what Croatia is and what a Croatia team is. We are a small country and won against one of the favourites at the World Cup.
"They are great fighters and only Croats can do that. Now we're in another semifinal and we shall not surrender. Croatia has shown something that not many teams can show."
Eight of Croatia's last nine knockout games at major tournaments have now gone to extra time -- a run stretching back to Euro 2008.
They have won only one game in Qatar in normal time but have booked their place in the semifinals having reached the final in 2018, where they were beaten by France.
"We have very strong character and don't give up," Dalic said.
"I believe only teams with great and strong character can come back and it's become normal for us. Strength of character is a part of our team.
"I am quite clear that people are being surprised by our performance for a second time in a row, it is such a phenomenal thing.
"I wanted to pass the group stage and take it one step at a time and we wanted to show patience. Now we are in the semifinals, it is not normal.
"We have confidence in ourselves. Croatia is a surprise indeed but this should have become less surprising."