Outstanding bowling from captain Will Sutherland, veteran Peter Siddle and fellow paceman David Moody gave the visitors a narrow lead on Sunday at the WACA.
Given the state of the other two current Shield games, and with no rain forecast in Perth for the next two days, whoever wins this match will most likely head to Adelaide for the final.
Second-placed Queensland are in dire straits against SA and third-placed NSW are also behind on the first innings against Tasmania.
Fourth-placed WA and Victoria, who started the final round in fifth, are behind Queensland only on points.
WA resumed on day two at 1-10 and their top order struggled as they chased Victoria's very gettable first innings of 197.
They slumped to 8-111 before Test debutant Cooper Connolly, batting at No.7, led a lower-order counterpunch that nearly gave them the lead.
But he was the last wicket to fall for a top score of 56 as the home side was dismissed for 186.
That set up an absorbing 18 overs at the end of the day, with WA desperate for quick wickets to wrest back the momentum.
Victorian openers Campbell Kellaway (20) and Marcus Harris (11) survived to stumps as the visitors ended on 0-32, a lead of 43.
WA paceman Joel Paris may well find himself on a dissent charge after he went very close to trapping Kellaway lbw with the first ball of the innings. The left armer glared at the umpire when his appeal was turned down.
Victorian wicketkeeper Peter Handscomb took seven catches as Sutherland, Siddle and Moody repeatedly asked searching questions of the WA batsmen.
Making his Victorian debut and playing against his old state, Moody claimed 4-41.
Sutherland took 3-32 as well as a great catch in slips, while Siddle's 13.2 overs yielded 3-10 and eight maidens.
Connolly combined with No.10 Cameron Gannon for a 53-run stand that gave the home side hope of a first-innings lead, but Moody had Gannon caught behind for 26.
After hitting eight fours and a six in his 104-ball knock, Connolly was unimpressed when given out caught behind off Siddle, with the ball cutting back and the umpire ruling there was an inside edge.
No.4 Jayden Goodwin made 36 as the only other WA batsman to reach 20.
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