Punt pulled the ball back for the experienced Forgesson to have a clear, and clean strike from six metres and she struck it just wide of the post.
Australia did get the ball in the net early in the second half but quick thinking by Forgesson, with captain Kayla Whitelock sitting out a two-minute suspension, who appealed to the umpires had the call reversed.
Australia had failed to take the free hit leading up to the goal with the ball moving instead of stationary.
Striker Olivia Merry got a one-on-one chance with Australia's goalkeeper Rachel Lynch in the eighth minute but Lynch blocked the shot.
New Zealand's one penalty corner - to two for the Aussies - had Lynch block Punt's low drive.
There was a pile of running and effort from both teams, with Whitelock and midfielder Stacey Michelsen particularly notable contributors, but ultimately frustration for the Black Sticks.
They will now play a classification match, either on Friday or Saturday, depending on their final placing in the pool.
New Zealand finished with two wins, two losses and a draw for seven points, while Australia ended with 10 points. Their hopes of finishing second in the group and qualify for the semifinals, depended on the world No 1 Dutch beating Korea in a game to be played early tomorrow.