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The central Americans won 1-0 in Qatar this morning (NZ time) to take the last available World Cup slot - but only after a contentious VAR decision ruled out a Kiwi equaliser and the New Zealanders also ended the game down to 10 men following the sending off of Kosta Barbarouses.
"If I’m being honest. I thought some of the officiating was absolutely atrocious," Hay said at the post-match news conference.
"The disallowed goal was two players battling for the ball. The foul could have gone to Matt Garbett to start with. Obviously VAR got involved and overturned that."
A check for a foul by the VAR decided Garbett had wrestled Oscar Duarte to the ground in the build-up to his cross that led to Chris Wood putting the ball into the net after 39 minutes.
"I haven't looked closely at the other one," Hay said of a second caution for Barbarouses, who was given a red card for a lunging tackle with some 20 minutes of the match to go.
"Our analyst, who was watching in the stands, wasn't convinced, but we could be proven wrong."
Hay says he was stunned by the appointment of Mohammed Abdulla, of the United Arab Emirates, to handle the one-off game.
"When we looked at what Australia got against Peru last night with good quality European officiating, I thought FIFA made a mistake in such an important game for us.
"I thought we were by far the better team. One team dominated, there was only one team trying to put together good quality football."
Wood said after the match he believed his side had created the most opportunities.
"We were the dominant side, possession stats would tell you, shooting stats would tell you.
"Ultimately it was fine margins. Little things went their way, and some decisions went against us which I think were very harsh."
The former Arsenal striker scored after three minutes, squeezing between two defenders to touch home a square pass from Jewison Bennette with the New Zealand defence slow to react.
His goal had some 6000 travelling supporters, out of a crowd of 10,803, on their feet but did not precipitate the expected dominant performance from the central American team.
Instead it was New Zealand who looked the more enterprising of the teams, with centre forward Wood, who was constantly watched by a cautious Costa Rican defence, the target of their attacks.
Alex Greive and Garbett also had chances in the first half but their shooting was off target.
Three changes by Costa Rica at halftime lifted them out of their slumber but it was still New Zealand who showed more attacking enterprise in the second half but without any return.
Costa Rica goalkeeper Kaylor Navas made a good stop in the 76th minute from Clayton Lewis as New Zealand toiled in search of an equaliser with Costa Rica sitting back and absorbing the pressure until the final whistle set off joyous celebrations.
The playoff between the fourth-placed team in the CONCACAF region and the Oceania confederation winner, at the Ahmad bin Ali Stadium, brought to an end the qualifying campaign for the 2022 finals in Qatar, which runs from November 21 to December 18.
Costa Rica qualified for a sixth World Cup in total and will compete in Group E at the finals against Germany, Japan and Spain.
New Zealand have been to just two World Cups, 1982 in Spain and 2010 in South Africa.
- additional reporting RNZ