The right-arm quick sprayed his hat-trick attempt, sending a delivery to Mitch Swepson wildly down the leg-side to miss the milestone.
The Bulls made 147 in their first innings when off-spinner Corey Rocchiccioli took a career-best 7-52.
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Rocchiccioli traps Bulls in web of spin with seven-for
WA, with Sam Fanning (95) falling just shy of his maiden first-class century, replied with 312 to take a stranglehold on the fixture.
Queensland's batters struggled again in their second innings with Ben McDermott and Lachlan Hearne top-scoring - both made 33.
Angus Lovell (26), his opening partner Matthew Renshaw (10) and Michael Neser (12) were the only other batsmen to reach double-figures.
WA's Morris collected 3-35 from 13 overs - he also removed McDermott with a brute of a short ball which reared at the batsman, who edged to wicketkeeper Joel Curtis.
Rocchiccioli (2-27) finished with his best first-class match figures of 9-79 while paceman Joel Paris (2-36) and Cameron Gannon (2-25) also took multiple second-innings wickets.
WA, the reigning champions who have won the Shield in the past three summers, entered the fixture at the foot of the ladder but are now within touching distance of the top two.
Queensland, who had held third spot, will likely slide to fifth.
Both states have two games remaining before the final, with WA to host NSW from March 6 and Victoria from March 15 at the WACA Ground in Perth.
Queensland will be on the road for their remaining fixtures, against Tasmania at Bellerive Oval and South Australia at Karen Rolton Oval on the same dates.