New Zealand play Fiji in what many had predicted to be Thursday night's gold medal game.
In the other quarters, France play Japan, Argentina meet Great Britain and Australia play South Africa.
In their last pool match against Great Britain, New Zealand slipped to a 21-0 halftime deficit before roaring back after the break with three tries.
Giant-slayer Japan's earlier defeat of Kenya meant New Zealand had to win to assure themselves of a quarter-final position.
Sam Dickson, who sparked the fightback against the Brits, said their performances to lose to both Japan and GB have been "really disappointing".
"We just hope luck goes our way. "We had a lot of confidence and faith coming into the Olympics.
"I don't want to make an excuse but geez we've had a bumpy year with injuries and all that and once again the first game we got her we lost two of our strike weapons (Sonny Bill Williams and Joe Webber)."
New Zealand also appear to have lost captain Scott Curry for the tournament as he played no part in the third pool game and was not suited up.
Dickson said it was hard to put a finger on why they have struggled.
"We just had a slow start in that game, and also yesterday against Japan we battled early, but it just shows that if we build the ball and build some phases we're good enough to put on points to any team," he said.