The Cubans will play on Thursday for a chance to win Pool B, while Australia faces Mexico on Wednesday in a rematch with both teams facing elimination.
Capping the comeback for Cuba was pinch-hitter Yosbany Peraza, who in the top of the eighth inning with two outs smacked a two-run, line-drive home run into left field.
"I knew I really needed to hit, hit strong," Peraza said, "because we were behind, we were not on top, and that's exactly what I hit, a strong winner."
But prior to Cuba landing the winning punch, it seemed as if it was Australia that would be making travel plans to San Diego for Round 2.
Australia held a 4-2 lead after six innings that all but nullified the effects of Cuban starter Albertin Chapman.
"Moments like those, we've lived them before," Cuba manager Higinio Velez said. "This is definitely one more. Boy, Australia has definitely given us a run for our money."
In a back-and-forth battle, Cuba took a 1-0 lead in the top of the third inning when Frederich Cepeda dropped a hit into shallow right field that brought Luis Nava home from third.
Australia, on the other hand, got its first hit in the third on James Beresford's double to left field. The inning would end, however, with Chapman striking out the side.
The lanky left-hander was impressive, hitting 97, 95 and twice 99 mph in one at-bat against Ben Risinger that ended actually better than it started.
Second baseman Hector Olivera backhanded a grounder hit by Risinger and flipped the ball out of his glove to shortstop Nava, who threw it to first for the out.
But Chapman slipped up in the fourth inning when he was he called for a balk. The miscue moved Luke Hughes to third after he led off the inning with a double, and Hughes scored on a single to left field by Risinger to tie the score at 1.
Chapman didn't return to the mound in the top of the fifth. In four innings, he allowed three hits and a run while striking out seven.
"In order to become the best pitcher, I still need lots of things," Chapman said. "I need to improve professionally. I need to work with curveballs. I need to work with different kinds of pitches."
Christopher Ramirez is a contributor to MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.