Timo Werner and Pape Sarr were both on target in the first half, with Matheus Nunes reducing the deficit on the stroke of half-time.
Spurs held on against a much-rotated City side, which suffered its first defeat of the season after a 2-1 reverse in north London.
Tottenham disappointed in a 1-0 loss away to Crystal Palace in the Premier League before this win, though Postecoglou was delighted with his side's response.
"It's a cup tie played against a formidable opponent, I thought we did well for the most part," he said. "We didn't allow too many clear-cut chances. We could have put the game away, but City are always going to push you.
"It's an ever-evolving process, this team is growing all the time, but when you have growth you are going to stumble.
"As disappointed as we were on Sunday I had great belief we could come out today and put up a great show.
"Both our goals were quality goals, they have to be as City don't give easy chances."
Spurs had been eliminated in its last three Carabao Cup ties against fellow Premier League sides, going out to Chelsea in 2021-22, Nottingham Forest in 2022-23 and Fulham in 2023-24.
Werner eased any nerves after just five minutes, however, with his close-range finish easing the mounting pressure on the former Chelsea forward.
"Great for Timo. He is going through a tough time, but the build-up play was great," Postecoglou said. "It should lift him, we are supporting him, but it is down to him and his approach to things.
"He was unlucky to not get one or two more, if attacking players don't score they feel anxiety a bit."
Postecoglou hammered his side after its collapse at the start of October in a 3-2 defeat to Brighton, and similar criticism came after after a defeat to Palace last time out.
"I don't believe in big moments or catastrophic moments - we are the same team that we were three days ago. Hopefully, we have more good days ahead of us," he said.