Wambach fired the Americans into an early lead at St James' Park, the home of Premier League club Newcastle United, on Friday as the holders booked their place in the last four.
But it was not until three minutes from time, when Sydney Leroux added a second goal, that victory was secured.
USA will now face Canada in the semi-finals. "This team is bound for glory," said Wambach. "It might not look pretty in every second of the game, but we get it done."
Wambach paid tribute to New Zealand, saying: "They are very physical. They're one of the most improved teams in this tournament in the last few years. You have to give them credit.
"The work-rate they have is amazing, but they couldn't save it in the second half as they got a bit tired. They have a fantastic future and they're a great team."