The Bushrangers enjoyed a good day with the bat at the MCG, posting a first-innings total of 450 in reply to the visitors' 243 to lead by an imposing 207 runs.
WA opened their second innings after tea with Wells (46 not out) and Bosisto (38 not out) appearing comfortable as they progressed the score to 0-86 at the close of play.
"Victoria were certainly on top of the game, but when we went out to bat all we could have asked for was to have 10 wickets in hand at stumps," Bosisto said.
"We'll look to bat sensibly and positively tomorrow and we'll see how the game plays out."
Earlier, a gutsy knock from Victoria wicketkeeper Seb Gotch helped the home side to a commanding position after they had resumed at 4-247.
Gotch was closing on a maiden first-class century when he spooned a leading edge to Marcus Stoinis off the bowling of Ashton Agar on 98.The Warriors were frustrated by a ninth-wicket partnership of 98 between Gotch and Scott Boland, who was unbeaten on 29 from 130 balls.
A brilliant 96 by Glenn Maxwell helped set up the innings, with Cameron White chiming in with an important 82.
Agar finished with figures of 4-119 from 50 overs in his return to first-class cricket after a broken finger suffered on Australia's tour of India.
Having won the toss and elected to bat, Bosisto top-scored in WA's first innings with 57 and wicketkeeper Josh Inglis made a handy 46, while Victorian spinners Fawad Ahmed and Jon Holland took three wickets each.
The Bushrangers are still searching for their first win of the season as they look to climb off the bottom of the Shield table before the Big Bash League break.