But while the hosts grabbed a 10-run first-innings lead, they were again left unimpressed by two controversial dismissals of their key batters, captain Roston Chase and white-ball skipper Shai Hope on Thursday (Friday AEST), which may have tipped the game in Australia's favour.
The two captains had proved the visitors' biggest obstacle as they put on 67 for the sixth wicket and didn't look in any great trouble as the Windies threatened to take a decent first-innings advantage, going to lunch at 5-135.
But Chase looked hugely frustrated early in the afternoon session when he was given out lbw to Cummins, convinced that he'd got an inside edge before the ball thudded into the pads.
The Ultraedge review gave no clear evidence one way or another, and the disgruntled captain had to depart for 44 off 108 balls.
Hope, back in his first Test for more than four years, had moved fairly trouble-free on to 48 off 91 balls when his inside edge off Beau Webster produced a brilliant one-handed diving catch from Alex Carey, only for the review to show that the ball may have touched the ground as the keeper thumped against the turf.
The third umpire, though, gave Carey the benefit of the doubt.
With their two key men gone, the West Indies were left simply hoping to battle their way to any first-innings lead, however slender, and paceman Alzarri Joseph did his bit to get them there, smacking an unbeaten 23 off 20 balls to put them marginally ahead.
But after they had only managed to grab a single wicket in the morning - Brandon King (26) shouldering arms and getting bowled by Josh Hazlewood - Australia gleaned five wickets fell in the second session for just 55 with Webster proving impressive in his six overs, finding pace, movement and bounce as he took 2-20.
The familiar triumvirate of Mitchell Starc (3-65), Hazlewood (2-34) and Cummins (2-34) all chipped in, while Nathan Lyon grabbed the final wicket of the innings, tempting Jayden Seales to slog one straight down Webster's throat in the deep.
Having started the day at 4-57, the West Indies otherwise showed real resolve as Hope reached 31no and Chase was unbeaten on 44 at the lunch break.