He also said "it's a shame" that the team will miss its inspirational player and captain Max Mata after being double- booked by referee John Pitti.
"It was the first game and both teams attacked well and the action was end to end. But looking back I would say we should have won it," said Hay after New Zealand came back from a first-half deficit to hold Turkey to a 1-1 draw.
"It's a huge loss for us and it was a tough decision in the 93d minute. It's a shame the World Cup will miss him in the next game against Paraguay. But it provides an opportunity to another player to step in," said the coach.
Hay said his players were initially rattled by the pace of the game but then got used to it.
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He conceded that the rain, under which the match was played, should have helped his team who are used to such underfoot conditions at home.
Turkey coach Mehmet Hacioglu blamed lapse in concentration of his defense in conceding the equaliser to a free-kick.
"I am disappointed that we conceded a goal because of our own mistake. In the training I always say to my players to look at the ball and don't turn your face. But it was a sudden lack of concentration and because of this we conceded the goal. I can say that we are still trying to adapt to the climate and the lack of concentration was caused by this reason," said the coach.
"I am not talking about individual performances. As a whole team, we could not play very well and in our own style and that's why the result was draw. It was not because of New Zealand's physical advantage," he said.
"I believe in players fully, so despite this result, we want to reach the target (of advancing to next stage)."
On the team's goal scorer Ahmed Kutucu, the coach said: "Ahmed is new to our team and comes from Germany where he plays for FC Schalke. Even if he has scored the goal, he is still a work in progress to reach the level which I expect.