Australia is a three-time Olympic champion, but its last gold medal performance was at home during the 2000 Sydney Olympics. New Zealand, meanwhile, is improving after failing to win a single match at the 2008 Beijing Games.
New Zealand had the ball in the Australian net in just the second minute, but the umpire had already blown for a short corner for an earlier offence. From the corner, Finlayson hammered in the rebound for her 18th international goal.
Australia should have equalized before the break but first veteran Megan Rivers hit the post with a low hard shot and then Emily Smith could only shoot straight at New Zealand goalkeeper Bianca Russell when all alone in the circle just before half time.
Australia threw everything into attack in the second half but could not break down the stubborn Kiwi defence and could have gone further behind had it not been for a great low save by goalkeeper Toni Cronk to turn away a Charlotte Harrison backhand shot.
"We created a lot of chances in the first half," said New Zealand captain Kayla Sharland.
"We scouted them a lot, we watched a lot of video footage of them," she added. "We knew our gameplan two months ago, we knew what we needed to do. We wanted to have a good start and get on a roll, I'm just very happy for the group."
The match was the first Olympic hockey match ever played on a bright blue pitch that is intended to make the fast-moving ball easier to follow for players, fans and television viewers.
Group B also features one of the tournament favourites, Argentina, along with Germany, South Africa and the United States. Defending champion the Netherlands is in Group A with Belgium, China, Britain, Japan and South Korea.
The men's hockey tournament was scheduled to start Monday.