The Kiwis won all of last weekend's four races and went 4-0 up with victory in race five on Saturday.
But defending champions the USA replied with their first win of the best-of-13 series in race six in Bermuda.
The American team began with a one-point advantage after winning the earlier qualifying regatta, which was erased by New Zealand's race-one win.
The final is a repeat of the last time the trophy was contested in 2013, when Team USA came from 8-1 down to win 9-8.
Four years ago, New Zealand were eventually overhauled by an Oracle team that featured Britain's Ben Ainslie as tactician in what is regarded as one of sport's great comebacks.
Ainslie's British team were knocked out at the semi-final stage this year, losing 5-2 to the Kiwis.
The America's Cup, the oldest competition in international sport, was first raced in 1851 around the Isle of Wight and has only been won by four nations.
Emirates Team New Zealand moved into a 4-1 lead over Oracle Team USA as racing resumed at the America's Cup.