The 21-year-old opener backed up the 125 he made against NSW at Brisbane last month with 103 at the WACA ground to set up the Bulls' first win in this year's competition.
WA are bottom of the ladder with three losses in three despite a captain's innings from Ashton Turner as he tried to fashion an unlikely win.
Turner was last out for 58 as WA were dismissed for 246 in 44 overs. Queensland had made 261 all out in 45.2.
After Turner put the visitors in, Weibgen overcame a difficult start in which Queensland stumbled to 2-37 after Jack Clayton edged behind for a duck in the second over, and Angus Lovell fell lbw to Kieran Elliott in the sixth.
But Weibgen and Hugo Burdon steadied for the Bulls, building a 112-run partnership for the third wicket. It ended in the 19th over when Burdon was caught off Esterhuysen for a well-crafted 67 off 49 balls, laced with seven fours and four sixes.
Four balls later, Esterhuysen made it 4-149 when he had Max Bryant caught behind for a duck.
Weibgen went on regardless with his 89-ball knock featuring 12 fours and two sixes.
After he departed at 6-210, caught off medium pacer Brody Couch, the Queensland tail swung a little to add 51 more.
Couch was the pick of the WA bowlers with 3-48, along with Elliot (3-57).
WA made a solid start to their chase before opener Mitch Marsh fell lbw to Tom Whitney for a 27-ball 36, at 1-58 in the eighth over.
Fellow opener Joel Curtis and Sam Whiteman then added 91 runs for the second wicket before Curtis was caught off quick Hayden Kerr (3-31) for 63 off 76 balls.
Whitney (2-41) had Whiteman caught behind for 40 before Will Malajczuk, Hilton Cartwright and Ashton Agar all fell cheaply, the latter two run out, reducing the hosts to a precarious 6-180 in the 31st over.
With 82 needed off 19 overs the run-rate wasn't an issue, staying in to get them was. Turner and Elliott added 38 but just as WA began to look favourites a clatter of wickets appeared to have ended their hopes.
After Elliott (15) was caught off Kerr, Couch and Esterhuysen swiftly followed as three wickets went down for two runs in 14 balls.
The rate remained feasible but with No.11 Jordan Quiggan joining him Turner knew he would have to get most of the remaining 42 runs himself,
The next 11 balls went for 26, including extras, during which the captain went to 50 off 51 balls and hit Mitch Swepson for six.
But his opposing skipper had the last laugh, dismissing Turner just as he seemed on course to secure an improbable win.
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