After recording a pair of ducks in the first round against Queensland, Bancroft was caught behind for eight shortly after lunch at the WACA on Monday.
It came on a rough day for Australia's Test aspirants, with Sam Konstas, Marcus Harris and Matt Renshaw all failing to make an impact in their Shield matches.
Konstas was lbw for two for NSW against Victoria at the MCG, before Harris fell for 16 in the second innings after being out for 26 in the first.
And Matt Renshaw was also lbw for two for Queensland against South Australia, as his hopes of returning to the Test team in the vacant spot at the top of the order continued to dim.
Those outs opened the door for Bancroft to ram home his case, after last week being selected in the Australia A squad to face India A alongside Harris and Konstas.
After Tasmania posted 277 batting first at the WACA, Bancroft opened his account for the summer by edging two consecutive boundaries through the slip and gully.
On his way, NSW opener Sam Konstas fell cheaply in the Sheffield Shield match against Victoria. (Morgan Hancock/AAP PHOTOS)
He was struck in the helmet on the next ball he faced from Riley Meredith, ducking into a short delivery from the quick.
The right-hander then fell shortly after, edging a ball outside off stump from Kieran Elliott through to wicketkeeper Jake Doran.
Bancroft has been the leading run-scorer in the Shield over the past two summers, and on history should be the front-runner to open against India in Perth on November 22.
But a lack of runs now threaten to let him down when it matters most, after also being caught behind for 0 in his opening two innings of the Shield earlier this month.
Fellow Western Australia opener Sam Whiteman was also caught behind shortly after Bancroft's dismissal, leaving the hosts 2-42 midway through day two.
Earlier, spinner Corey Rocchiccioli was the main destroyer for West Australians, taking 4-70.
The offspinner had Tasmania's leading run-scorer Mitchell Owen lbw for 69, after the right-hander had previously slog-swept him for two sixes over mid wicket.
Doran also hit 66 for Tasmania, before he became one of paceman Brody Couch's three victims for the innings when he edged behind early on day two.
Mitch Marsh did not bowl any of the 87.3 overs for Western Australia, but is confident he will be able to fill the void left by Cameron Green's back injury with the ball this summer.
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