A five-wicket haul from evergreen paceman Peter Siddle and a brave batting performance lifted Tasmania to the thrilling victory.
WA began the day with the intention of extending the run chase for Tasmania, but those hopes began to fade as skipper Shaun Marsh was dismissed by Tasmania substitute Riley Meredith on 58.
WA lost 5-26 early on day 4 to be in all sorts of trouble at 7-155, with a lead of just 141.
A valiant partnership between Jhye Richardson (50* off 59 balls) and Joel Paris (27 off 69 balls) helped increase the run target.
Richardson's fourth half-century of his first-class career featured eight boundaries.
However, their time at the crease came to an end after 72 runs between the pair, as Beau Webster snared Paris.
WA set Tasmania the task of scoring 217 to win, Charlie Wakim (50 off 81 balls) and Caleb Jewell (52 off 51 balls) produced handy knocks to set up the successful run chase, while Jarrod Freeman (21no) and Jake Doran (19) held their nerve after Tasmania had fallen to 6-186.
Siddle hit the winning runs after Doran was dismissed late, helping the Tigers reach the victory target with 26 balls to spare as rain clouds hovered close to the Ground.
Richardson was forced to sit out days two and three after suffering a back spasm while warming up on Monday.
But he didn't show any signs of soreness during his blistering knock, and he returned figures of 2-37 off 14 overs with the ball.
"It was a fantastic game of cricket," Richardson said.
"It was disappointing we couldn't get over the line, but a wicket like that produces a result and that's what we want.
"I found looking to score was probably the way to go about it instead of shutting up shop.
"The idea was to go out there and try to hit some quick runs, but that's the way I like to go about it anyway. "