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Jerry Tuwai's last-gasp try was enough to see Fiji home after the Kiwis had scored twice in the first spell to lead 10-7 at half-time.
Vatemo Ravouvou had opened the scoring for Fiji after two minutes, set up by a perfect offload from Amenoni Nasilasila.
New Zealand responded through Vilimoni Koroi two minutes later, then grabbed the lead right on half-time when a rolling maul barrelled through for Sione Molia to touch down.
The scoreline stalled at 10-7 after the restart until Tuwai burst through the flagging Kiwi defence nearly two minutes after the final hooter had sounded.
"It was frustrating and disappointing," Laidlaw said afterwards.
"I thought tactically we were really smart and executed our game plan, and we thought we'd done enough to win the game."
The New Zealanders had earlier begun their Las Vegas campaign with a 28-7 win over Uruguay and surprise 26-19 defeat to Argentina.
It was just the 10th time in 82 matches that the Kiwis have lost to Argentina, who scored late to secure the win.
The All Blacks Sevens went on to win their third pool match against Scotland 26-17 on Sunday morning.
With the Commonwealth Games looming, Laidlaw says his team have to find ways to clinically close out tight games.
"We've grown, we're getting better and we're stronger every time we play," he said.
"Yesterday wasn't good enough - the way we played was frustrating. So today we came back and we've been better, but again just not quite enough."
New Zealand are currently second on the World Sevens Series ladder behind South Africa, having won just one 2017-18 event in Cape Town.
The All Blacks Sevens are just seven points ahead of third-placed Fiji, who won last month's Hamilton event.