Van Persie scored twice in the Netherlands' shock 5-1 win against world and European champions Spain in Salvador on Friday.
Spain were made to look amateurish at times against a rampant Netherlands and as a result, expectations have risen significantly for Louis van Gaal's side.
But Van Persie - who acknowledged the result had changed perceptions of their hopes in Brazil - is urging his team-mates to not get carried away.
"The tournament has barely started yet," Van Persie told FIFA.com.
"We achieved an impressive result but we have a long way to go to win this tournament.
"This is my fifth (major) tournament and I know how these things work - the euphoria vanishes just as quickly as it appears.
"So we have to make sure that we hold on to it - not only for the people and fans, but also for ourselves.
"Everybody is euphoric and happy in the Netherlands. We are dealing with people's expectations here, which had been low compared to other years.
"I actually think this worked to our advantage because nobody expected much from us. But after such a performance, the dynamics have naturally changed.
"However, as a country, we shouldn't get ahead of ourselves. We have to stay realistic."
Van Persie etched his name in World Cup folklore with an outstanding diving header in Friday's match that drew the Netherlands level after Xabi Alonso's penalty had given Spain the lead.
His dive was mimicked by elated fans all over the world on social media platform Twitter, who posted photos of themselves trying to imitate the striker.
"I've seen the spectacular footage and images from the celebrations across the world, and that makes me so proud," he added.
But the forward has quickly turned his attention to Wednesday's meeting with Australia in Porto Alegre.
After a poor first 15 minutes, Australia recovered well and regularly troubled Chile in their 3-1 defeat on Friday.
And Van Persie knows a change in style is needed ahead of their meeting against the Socceroos.
"Now the focus is on Australia, and I think that will be completely different to the game against Spain," he continued.
"Australia have a very different style and I think we are going to have to adjust our own game to beat them. It's an interesting process, having to adjust your game every time.
"However, we have fantastic technical staff who know exactly how to prepare us for matches, so I'm not worried."
