An in-form Josh Inglis (101 off 141 balls) put the hosts in control on day three at the WACA Ground, before a batting collapse left the Tasmanians' chances of victory in disarray.
When bad light stopped play, Tasmania (9-135) were only 80 runs ahead of WA with one wicket in hand, having sputtered to 3-13 before losing 6-67 in the final session.
Matthew Kuhnemann (5) and Riley Meredith (2) will start unbeaten at the crease on day four, but neither has recorded a first-class half-century in their career.
The stage appears set for Bancroft and fellow opener Sam Whiteman to bat WA home in reply, as the former chases the vacancy at the top of the order for the Test summer.
A Test player on 10 previous occasions, Bancroft managed only eight runs in the first innings, but could finish the week as the most in-form player of those in the mix.
Neither of Bancroft's two clearest rivals for the opening spot in the series against India scored above 50 in Victoria's ongoing clash with NSW, Marcus Harris and Sam Konstas both misfiring.
Unbeaten on 72 after day two, Inglis made his way into the 90s on Tuesday with a cut shot for four off Beau Webster (2-74), before losing partner-in-crime Cooper Connolly (55).
Josh Inglis scored a century to put Western Australia in control of the Shield clash with Tasmania. (Richard Wainwright/AAP PHOTOS)
The WA young gun had continued his strong start to the summer by bringing up his half-century just after the rain-delayed start to play on day three.
But he was caught by wicketkeeper Jake Doran off Webster to bring a 161-stand with Inglis to a close.
On 99, Inglis overcame Connolly's departure to reach triple figures with a single to deep mid-wicket off former Australian white-ball teammate Matt Kuhnemann (1-73).
But Inglis faced only three more deliveries before falling victim to the same duo as Connolly, his outside edge from Webster's bowling caught safely by Doran. WA were all out for 332.
Tasmania threatened to implode, slipping to 3-13 inside five overs as veteran quick Cameron Gannon dispatched of dangerous opener Caleb Jewell (5) and captain Jordan Silk (0) in the space of five deliveries.
After he helped stem the bleeding, Doran (34) was caught by Inglis on the second delivery after tea as pressure mounted on the visitors.
WA paceman Brody Couch had both Tasmania's top-scorer Charlie Wakim (37) and Brad Hope (0) caught in the space of three balls, and when Webster went for 35, Tasmania were in a world of hurt.
Earlier, match officials permitted Hilton Cartwright to return to the crease after the WA allrounder retired on day two to attend the birth of his child.
But Cartwright added only 13 more runs before holing out to Bradley Hope with a pull shot to the deep on 65 runs.
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