Fury skipper Chloe Piparo won the toss and sent the league leaders into bat, and the Tigers posted 192 runs for the first three wickets.
Georgia Redmayne (77) and Corinne Hall (51) looked to be cruising, until the West Aussie bowlers wrestled back some momentum.
They took 7-50 in the final ten overs of the first innings to restrict Tasmania to 9-242.
Kath Hempenstall (3-39) was the pick of the bowlers, while Kate Cross (2-33) and Emma King (2-41) both returned good figures from their ten overs.
The Fury's run chase didn't start the way they would've liked with opener Bhavi Devchand trapped LBW on the first ball of WA's batting innings.
Inform all-rounder Heather Graham (10) couldn't add to her string of half centuries, trapped LBW by Alex Hartley.
The West Aussies looked down and out at 7-83 half way through their innings, before Amy Edgar (67) and Emma King (27) combined for a 68 run partnership to breathe some life back into the Fury chase.
The fightback couldn't be sustained, and WA were all out for 164 runs, 78 short of victory.
WA's finals chances are now dashed, the Fury can finish their season on a high with a win against the third placed Queensland Fire on Sunday at Casey Fields.