Smith produced the standout performance of the evening, averaging 99.81 and converting 86% of his attempts at double in a 6-2 victory over third seed Stephen Bunting.
The Australian was ruthless when it mattered, following finishes of 92 and 90 with a superb 124 on the bull to seal his place in the last eight.
“It feels pretty awesome. Eight years since I last played on a PDC stage, so I’m pretty stoked,” said Smith.
“I think I played really well. What we’re doing in the ADA, DPNZ and ANZ Premier League is setting us up for this sort of stuff.
“What the PDC is investing in the Oceanic region is showing in the results.”
Leek also marked his World Series of Darts debut with a dramatic 6-5 victory over Ross Smith, surviving six match darts before taking the deciding leg.
The Englishman had led 4-1, but Leek won five of the final six legs as Smith struggled on the doubles.
“It’s the best feeling in the world. I’m lost for words,” said Leek.
“I tried to compose myself as much as I could and thankfully that last dart went in.
“It’s a dream come true to play on this stage, so I’ll make sure I’m ready for the next one and we’ll see how we go.”
Elsewhere, top seed Gian van Veen impressed on his first appearance in New Zealand, averaging more than 97 in a 6-1 win over Jonny Tata.
The Dutchman will face Jonny Clayton in the quarter-finals after the 51-year-old recovered from 4-2 down to beat home favourite Haupai Puha.
Fellow Welshman Gerwyn Price will meet Damon Heta after beating Mark Cleaver 6-1, converting 67% of his attempts at double.
Heta, meanwhile, overturned Ben Robb with a burst of five legs on the bounce, following 11- and 12-dart legs with a 105 finish.
Josh Rock and James Wade will complete the quarter-final line-up, owing to their triumphs over Simon Whitlock and Kayden Milne.
Rock held off a Whitlock comeback to win 6-4, while Wade marked his return to New Zealand with a 6-0 victory over Milne.
The NZ Darts Masters concludes at Spark Arena on Saturday, with the quarter-finals, semi-finals and final all taking place during the evening session.
Van Veen and Clayton will open the quarters, before Wade faces Rock, Price takes on Heta and Smith meets Leek.
Results
Gerwyn Price 6-1 Mark Cleaver
Adam Leek 6-5 Ross Smith
James Wade 6-0 Kayden Milne
Josh Rock 6-4 Simon Whitlock
Gian van Veen 6-1 Jonny Tata
Jonny Clayton 6-5 Haupai Puha
Damon Heta 6-4 Ben Robb
Raymond Smith 6-2 Stephen Bunting