Klinge was the only Oceanic player to progress from Friday's action in Wollongong, producing an 11-dart break in the deciding leg to defeat Smith.
Smith hit eight 180s and averaged 102.88, but missed two match darts in the final leg as Klinge took his chance to secure the biggest win of his career.
“I’m over the moon with that. That was the adrenaline rush of my life,” said the 34-year-old, who averaged 96.
“I felt a lot more comfortable playing this time around, and it showed in my performance, I think.
“Ross was throwing everything at me. I was lucky he missed at the end there, but I took my chance.
“I’m on a free rein now. I’m just going to do my best and keep going. There’s no pressure on me.”
Australia's Klinge will face Gian van Veen in the last eight after the world number two beat Raymond Smith 6-3.
Van Veen sealed his victory in style, producing a superb 161 checkout to finish off a tidy performance.
Wade then recorded the highest average of the day, posting 112.73 in a 6-0 battering of Tim Pusey.
His average was the second-highest recorded at a World Series of Darts event in Australia. He finished with a sensational 164 checkout on the bull.
“That felt amazing. Everything just fell into my hands there,” Wade said.
“I think I’m guilty of sometimes forgetting how lucky I am to do what I do, but I’m just happy to be here, and I’m playing alright.
“I’m not here to make the numbers up, but I’m amongst some great players. If I replicate that level tomorrow, everyone is stuffed, but tomorrow is a different day!”
Wade will meet Stephen Bunting in the quarter-finals after the Liverpudlian beat Mal Cuming 6-3.
Bunting won the opening three legs before Cuming threatened to fight back, but 'The Bullet' held on to progress.
Gerwyn Price also moved into the last eight with a 6-2 victory over Australian fan favourite Simon Whitlock. The Welsh star won four successive legs and finished with back-to-back checkouts of 124 and 108.
Price will face Josh Rock, who also came from behind to beat Adam Leek 6-3. Rock trailed 3-2 before winning four straight legs and producing two ton-plus checkouts.
Jonny Clayton, meanwhile, began his bid for back-to-back World Series titles with a 6-3 triumph over Ben Robb. He will face Australian number one Damon Heta, who averaged 102.39 in a 6-2 win against compatriot Joe Comito.
The Australian Darts Masters concludes on Saturday, with the quarter-finals, semi-finals and final all taking place at the WIN Entertainment Centre.
Results
Brody Klinge 6-5 Ross Smith
Stephen Bunting 6-3 Mal Cuming
Josh Rock 6-3 Adam Leek
James Wade 6-0 Tim Pusey
Gian van Veen 6-3 Raymond Smith
Jonny Clayton 6-3 Ben Robb
Gerwyn Price 6-2 Simon Whitlock
Damon Heta 6-2 Joe Comito