Barisic started his first match of the tournament in place of the ill Borna Sosa and played the full 120 minutes alongside Juranovic as the 2018 runners-up claimed a penalty shoot-out win over Japan following a 1-1 draw.
Standing between the Old Firm pair and a place in the semi-finals in Qatar are the tournament favourites, who thrashed South Korea 4-1 in last night's other last 16 tie.
But Barisic, who had been an unused substitute throughout the group stages, insists Brazil will hold no fears for his side ahead of Friday's last eight fixture.
"We are in the quarter-finals, we showed character, that we have the width of the squad," Barisic told Croatian media.
"Japan is powerful in racing, it was not easy to break them. But our passage is deserved. This match confirmed that every player is important, that we are all here for one goal. It is a good result and to make our fans happy. The next opponent? We are ready for anyone, we have the quality to play with anyone."
There was dejection for another Old Firm star as Celtic's Daizen Maeda was left on the losing side despite scoring the opener in the 43rd minute, reacting quickly to slam home from close range after a cross fell into his path.
Croatia equalised after the break with a header from Tottenham winger Ivan Perisic with Maeda replaced ten minutes later after delivering a performance that left the BBC Match of the Day studio guests signing his praises.
"He's been an absolute livewire. I think it's only his second international goal," said former England defender Rio Ferdinand. "He's looked like a man possessed up there. You wouldn't want to play against that. He’s running around, making it difficult for you. With and without the ball he's an absolute nuisance.
"In the first minute of the game he's closed the keeper down. You know you're in for it now. That's the catalyst for the others to come from behind. What I like about him, he's always on the move."