Just as in the previous round against Wolves, City were taken to extra time and then a nerve-shredding penalty shoot-out after a hard-fought quarter-final finished all square 1-1 with Bernardo Silva's first-half opener cancelled out by Jamie Vardy's 96th minute penalty.
And just as he did against Wolves, Bravo rose to the occasion, producing a superb save from Riyad Mahrez to secure a dramatic 4-3 shoot-out success for City after Vardy had sent his earlier sudden death spot-kick wide.
Zinchenko said: "I was due to be the fifth penalty taker but I am so happy because Claudio was the hero again and many congratulations to him and all the team.
"It was an amazing difficult game and Leicester were the champions just two years ago so we are very happy to get the win.
"We played not bad until the penalty and controlled the game until the last moment."
Goalscorer Bernardo, who produced a man-of-the-match display, echoed the salute to City's goalkeeper and admitted the side were delighted to have moved a step closer to Wembley.
"It was a tough game and we knew they had a very good team and it's very tough to play here but fortunately at the end Claudio gave us again the win and we are very happy.
"It has been a great first half of the season for us and I hope we can keep doing our job and winning our games as this team deserves to keep winning because of the way we are playing. We will try to celebrate something at the end of the season."