Turkey and Romania's wins took them ahead of Hungary, level on 16 points.
In next week's final matches, Turkey faces a tough match against the Netherlands in Istanbul, Romania hosts Estonia and Hungary has a home game against Andorra.
Van Persie scored two first-half headers and tapped in his third in the 53rd to bring his tally to 41 goals in 80 internationals. Goals from Jeremain Lens, Kevin Strootman, Rafael van der Vaart, Arjen Robben and a Szilard Devecseri own goal completed the Netherlands's scoring. Balazs Dzsudzsak netted a penalty for Hungary.
Van Persie's second which matched Patrick Kluivert's 10-year-old mark of 40 international goals was the pick of the night's strikes. Robben fed Lens deep in the Dutch half, Lens sprinted forward before backheeling the ball back to Robben, who crossed for Van Persie to head powerfully past Adam Bogdan in the Hungary goal.
Van Persie ran to the sideline where he jumped into the arms of Kluivert, who is now one of coach Louis van Gaal's assistants.
"Of course, it was a great performance," Van Gaal said of Van Persie's hat trick. "But it was also all about how we put Van Persie in a position to score."
Van Persie, who had been in doubt for the match with an injured toe, lauded Robben's contribution.
"I thought the second was fantastic. The pass (Robben) gave was so beautiful a floater I just had to nod in," Van Persie said. "The third was also a fantastic assist from Robben."