Defending 244 (a target that had been adjusted on the Duckworth Lewis Stern method), the home side needed 14 off the final six balls before death-bowling specialist Andrew Tye held his nerve with a clutch last over.
Tye took two wickets and conceded just six runs to see WA secure a tense eight-run win.
Shaun Marsh earlier put on a batting clinic on the way to an 84-ball 82, firing the visitors to 6-210 from their 35 overs after mid-afternoon rain ensured a lengthy delay in play during the WA innings.
Marsh was supported well by brother Mitch (30) as the pair combined for a 60 run partnership, while D'Arcy Short (35) also contributed well at the top of the order.
NSW looked in control of the run chase, before WA put the clamps on the Blues in the final overs.
Jhye Richardson (2-36), Nathan Coulter-Nile (3-47) and Tye (2-49) all stood up to help secure WA's third straight win.
Western Australia sit on top of the Marsh One-Day Cup table after three matches, and take on New South Wales again at Drummoyne on Wednesday.