Cuba are traditional heavyweights in the sport but they have never won the World Baseball Classic and have underperformed since reaching the final of the inaugural tournament in 2006.
The Cubans lost their first two games at this year's competition in the group stage in Taiwan but they won their next two and have now punched their ticket to Miami for the semi-finals.
Manager Armando Johnson said their quarter-final win would be celebrated long and hard back home.
"The Cuban fans are so emotional right now - they've been waiting for this moment for such a long time," he said.
"Everybody loves baseball so much and they were rooting for us. This is a new chapter for us."
Cuba included players from Major League Baseball in their squad this year for the first time and Chicago White Sox pair Luis Robert Jr (25) and Yoan Moncada (27) were both in the starting line-up.
Johnson said the process had not been smooth, with some MLB players turning down Cuba's invitations and others joining them at the last minute.
Outfielder Yoelkis Guibert (28), who plays independent league baseball in Canada, said "the team got stronger" as a result of the fresh injection of talent.
"I'm not saying that they did a great job right from the start, but everyone is of a very solid quality and they lift us up," he said.
Cuba broke open a tight game with three runs in the fifth inning but had to fend off an Australian fightback to claim the victory.
Baseball has been played in Australia since the 1850s but it enjoys only a fraction of the popularity of cricket, rugby or Australian rules football in the sport-loving country.
Manager Dave Nilsson said his team's exploits had given Australian baseball "hope for the future".
"We have a lot of belief as a country now and we know what we're capable of," he said.
"We know that we can continue to improve and play on the big stage."