The Manchester United striker scored the second goal in Wednesday's clash in Porto Alegre, equalizing for Louis van Gaal's men before Memphis Depay's eventual winner.
Van Persie, who picked up a suspension with a second yellow card of the World Cup which rules him out of the final group game with Chile, was also swift to praise the Socceroos' underdog performance.
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"The first half was very bad from our side," he told NOS. "But it says a lot if you strike back like we did. Australia just kept running and running.
"They played very well. At a World Cup you can't play every match perfect from the beginning until the end. Games like this you must win. In the second half we played our more familiar [4-3-3] system, which we are more used to.
"I'm very happy with this result, especially because I can't play against Chile. That's a real shame."
Australia had a golden chance to take a 3-2 lead before the Dutch capitalized with a winner of their own and Van Persie said such fine margins are decisive at the top level.
"Welcome to the World Cup," the Man Utd man said, regarding the topsy turvy nature of the match. "This is the biggest tournament in the world, so you know it's going to be hard. Even against Australia.
"We must enjoy this. We're almost in to the next round, nobody expected this."
Van Persie now sits joint-top of the goal-scoring charts, alongside teammate Arjen Robben and Germany attacker Thomas Muller, with three strikes.