The Kiwis came out firing in the first quarter and it was Hayward who got the Black Sticks on the scoreboard in the fourth minute. She calmly took a reverse stick shot from the top of the circle which went high to the keeper's right.
Haywards's goal was backed up by opportunities from Flynn and Olivia Merry, who were both given a chance to extend the lead, but it wasn't to be.
The second quarter wasn't as compelling for the Kiwis, New Zealand were called on to defend Korea's quick counter-attack, Rutherford superbly saved a penalty corner, and Korea's best chance ricocheted off the goal post.
In the third quarter the Black Sticks were back in the game, Kirsten Pearce was all smiles when she received a pass from Olivia Merry and low to the ground she pushed it over the line to double the lead.
In the final spell, the Kiwis executed a clinical penalty corner - captain Anita Punt flicked to Flynn who was waiting on the post to deflect it in and give her side a 3-0 lead.
Korea's Jung Eun Seo scored a field goal with two minutes left to play.
The Black Sticks Women play their next game against the Netherlands at 8.30am tomorrow, with the game LIVE on Sky Sport 2. There will be a replay of today's game on Sky Sport 2 at 1.30pm this afternoon.