Teenage opener Sam Konstas fell in single figures for the second time in a tough return Test and Cameron Green (15) continued to struggle at No.3 as Australia went to stumps on day two at 4-92.
The tourists lead by 82 runs after the visiting pacemen restricted West Indies to a first-innings total of 190 in reply to 180.
The ongoing 27-run partnership between a newly patient Travis Head (13 off 37 balls) and Beau Webster (19 off 24) appears critical for Australia.
"Hopefully that can continue in the morning and we can push it as far as we can, some low-order runs, and go from there," said Starc.
There's been plenty in the pitch for the bowlers as 24 wickets fell across the first two days of the three-match series - Australia's final tune-up before the home Ashes.
Australia's new-look top four managed a combined 47 runs in the second dig as the team fight to regroup following Marnus Labuschagne's axing and Steve Smith's finger injury.
"Obviously it's a different team without Steve and Marnus in it for the first time in a while, Steve I think is the only batter in that group that's played in these (Caribbean) conditions before," said Starc, Australia's best bowler with figures of 3-65.
"I think he's back in the next couple of days so that will be good around the group.
"It's been a challenge for these first few days and they've been working as hard as ever. It hasn't quite worked so far for a few of them but that's the beauty of Test cricket, it's a challenge."
Konstas's highly anticipated comeback Test will finish with scores of three and five, but he was fortunate to score at all on day two.
The talented teenager was dropped twice from the bowling of Shamar Joseph (1-15) in the space of three balls during the second over before getting off the mark.
Australia's familiar tormentor Joseph finally had his man when Konstas chopped on, the ball ricocheting from the toe of his bat and onto middle stump.
Green could also count himself lucky not to have gone sooner than 15 - still his highest score from four digs since being bumped up to No.3.
The towering West Australian successfully overturned the umpire's lbw call on 13, and added only one run before surviving Justin Greaves' review for lbw.
For the fourth time in as many Test innings since returning from a back injury, Green edged to the slips cordon after an uncomfortable stay at the crease - this time falling to Greaves.
Smith's injury replacement Josh Inglis (12) went cheaply for a second time this Test, shouldering arms and allowing Jayden Seales (1-24) to nudge off-stump with a peach.
Usman Khawaja's luck from the first innings also ran dry on 15, with ball-tracker inconclusive as the veteran sought to overturn the on-field call of lbw from Alzarri Joseph (1-16).
Earlier, West Indies' Roston Chase (44) survived an Australian lbw appeal on one and a Konstas dropped catch on four to forge a 67-run stand with Shai Hope (48).
But Chase looked hugely frustrated early in the afternoon session when he was given out lbw to Pat Cummins (2-34), convinced that he'd got an inside edge.
The Ultraedge review gave no clear evidence and the disgruntled captain had to go.
Hope would have been similarly disappointed to have been given out when replays cast doubt as to whether the ball had grazed the turf on its way to Alex Carey from Webster's bowling (2-20).
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