Victory put Costa Rica into their third straight World Cup and sixth in total. Their best effort so far was a surprise run to the quarter-finals in 2014.
New Zealand striker Chris Wood had what he thought was a leveller ruled out by VAR shortly before half-time at the Ahmad bin Ali Stadium due to a foul earlier in the move.
The Kiwis also saw Kosta Barbarouses sent off, barely eight minutes after coming on as a substitute, for a challenge on Costa Rica's Francisco Calvo. That decision was also changed by VAR after he had initially been shown a yellow card.
"We are incredibly disappointed, we are hurting," said New Zealand coach Danny Hay.
"I thought we were by far the better team. There was one team that dominated."
He called the decision for the disallowed goal "atrocious" and said the referee should not have been in charge of the game.
Both sides had suffered badly from the pandemic upheaval.
The whole Oceania qualifying group, that New Zealand won to reach this play-off, was held behind closed doors in Qatar this year because of the coronavirus.
Costa Rica star Bryan Ruiz went down with Covid-19 two weeks before Tuesday's game and the team had to deny accusations that they played two infected players against Jamaica in January.
New Zealand's neighbours Australia took the 31st qualifying spot in the same stadium on Monday when they beat Peru 5-4 on penalties.
The first World Cup in an Arab nation starts in Qatar on November 21, with the final on 18 December.