Billingsley scored before halftime then set up Monty Patterson and Joel Stevens after the break before Sam Brotherton and Clayton Lewis sealed the victory to give Oceania, FIFA's weakest confederation, the opportunity of two sides advancing to the round of 16.
Tournament debutantes Fiji had provided a massive upset with a 3-0 win over Honduras in Group F on Thursday and could also conceivably advance to the knockout stage after the final matches.
New Zealand had to battle back after Myanmar's Aung Thu scored a 28th minute goal before Billingsley equalised in the 40th.
Patterson added the second goal seven minutes after the break before Stevens, Brotherton and Lewis put the game beyond doubt in the final 25 minutes.
New Zealand are third in Group A on four points.
Ukraine won the group with seven points after a 3-0 win over the United States in Auckland. The Americans finished second on six points.
Argentina, the pre-tournament favourites drew their final game 0-0 against Austria in Wellington.
The Latin American side finished Group B with just two points following a draw with Panama and loss to Ghana, who won the group with seven points after they beat the Central Americans 1-0 in Auckland.
Austria finished second in the group on five points, with Argentina third.
Humberto Grondona's side have no chance of qualifying as one of the four best-placed third teams who also advance to the round of 16 along with each of the group winners and runners-up.
Four other third-placed teams, including New Zealand and Fiji, already have more points than Argentina.
Argentina's Giovanni Simeone had several chances in the first half to ensure his side advanced but was denied by some superb saves from Tino Casali.
The River Plate striker also unintentionally blocked a goal-bound shot by Nicolas Tripichio in the 10th minute.
In Auckland, Emmanuel Boateng's 74th minute goal was enough to send the 2009 champions Ghana through to the knockout phase.
(Editing by Sudipto Ganguly)
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