Needing a victory to advance to the knockout stage, Australia used a five-goal second quarter to grab an 8-5 lead by halftime, then scored the first three goals after the break to extend it to 11-5. Greece, which could have advanced with just a draw, never recovered.
"It was a relief. We had a lot of criticism, and we had a long meeting yesterday that was player driven so the accolades go to those guys," Australia coach John Fox said. "They said, 'look this is what we need to do to win,' and I was basically there steering and they did all the work.
"This Olympics is pressure like no tournament, so if something starts to break down you have to address it straight away. You have to have that honesty session and address those right away."
A bit of that pressure is off now for Australia with Monday's win putting the team into the quarterfinals as the fourth seed out of Group A. It will play the winner of Group B in the knockout stage.
Greece fell to fifth in Group A with the loss and was eliminated from the tournament.
"I believe we are a better team than Australia but congratulations to them. They showed a lot and deserve to go (through)," Greece's Christos Afroudakis said. "Sometimes there are surprises."
Group A leader Croatia had little trouble with winless Kazakhstan, pouring it on early to open up a 5-0 lead and cruise to an easy win. Samir Barac and Miho Boskovic both scored twice for Croatia, which already had secured the group's top seed for the knockout stage.
"It don't mean nothing before the quarterfinals," said Dobud, whose five goals Monday brought his tournament goal tally to 12. "If we lose there we go home, so we need to be good at the right time in the quarterfinal and then we go for medals, and that's the reason we came here."
With the loss, Kazakhstan goes home at the bottom of Group A.
While Croatia, which won the FINA World League title last month, emerges from the preliminary stage unbeaten and largely untested, it faces a much harder road in a knockout stage where it will have to tangle with four powerful teams advancing from Group B: gold medal-favourite Serbia, three-time defending champion Hungary, 2012 European championship silver medallist Montenegro and 2008 Olympic silver medallist the U.S.
Those four teams have all secured spots in the quarterfinals out of Group B, but their seeds for the knockout stage are still up in the air.
Serbia can secure the top spot with a win later Monday against Romania. The U.S. can wrap up no worse than the No. 2 seed with a win against Hungary, while a loss for the Americans will likely drop them to the fourth seed.
Montenegro, meanwhile, closes out the preliminary round against winless Britain, which only qualified for the tournament as the host nation.
Ryan Lucas can be reached on twitter at www.twitter.com/relucasz