The challengers started the regatta with a one-point deficit thanks to Team USA's topping the qualifying round-robin.
New Zealand last won the America's Cup in 2003 but it has lost two finals since then.
The first race in the contest to win the world's oldest sporting trophy was fought out on Bermuda's Great Sound by Oracle Team USA, skippered by Jimmy Spithill and Emirates Team New Zealand, helmed by Peter Burling.
Although they have won three races, the Kiwis lead Oracle 2-0. Oracle won a bonus point for winning the qualifiers, but it was actually a negative point for the Kiwis.
New Zealand are closing in on winning the America's Cup, having claimed four of the eight races they need to recapture the world's oldest worldwide sporting competition.
Spithill's Oracle team now face an uphill battle to win the 35th edition of the competition, where the trophy is earned by the first team to reach seven points.
The Kiwis continued their dominant form by winning both races in Bermuda on Sunday, adding to their two victories from race one and two on Saturday.
America's Cup holder Oracle Team USA is staring down the barrel of defeat against a powerful challenge from Emirates Team New Zealand.
"We're under no illusions that we've got a lot of hard work going forward", said Burling".
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Oracle does have history on its side.
The New Zealand team's double victory erased a one-point advantage the US team earned in an earlier regatta and gave them an early lead as they showed superior speed over the defending champions in the opening races.
Coutts has been the CEO of Oracle Team USA since it beat Alinghi in 2010.
The low-key Kiwi crew shook hands after crossing the finish line just off the America's Cup Village on the site of a former Royal Navy base. The team is hungry and a very, very competitive group.
"Clearly, now we need to put everything back on the table".
"Today was a really good day for us", Burling said.
"We feel like were far from flawless today, and we've got a lot to improve on in the next few days". We tidied up a lot of those little errors we made yesterday around the course and I think that really showed. We're just trying to keep learning and keep improving.
So far, the Kiwis and their cycling grinding system have proved too fast for Oracle in light, shifty wind.
But Spithill insisted there was still room to make useful adjustments to the complicated systems that control the space-age craft, and to "appendages" including the massive fixed-wing sails, rudders and the daggerboards that allow the boats to foil - virtually flying with their twin hulls above the water surface. The Kiwis used the same foils they did in speeding to two victories on Saturday while the American-backed crew appeared to use two different foils.