Australia had lost their last seven first-innings wickets for just 68 runs on day one, but West Indies suffered a similar fate on Sunday as they slipped from 82-3 to concede a first-innings deficit of 82.
Scott Boland, picked as an additional seamer in this Test in place of spinner Nathan Lyon, took 3-34 while Josh Hazlewood and Pat Cummins contributed with two wickets apiece.
John Campbell (36) top-scored but was one of only two West Indies batters to pass 20 before he was trapped lbw by Boland without playing a shot.
Australia then looked to build a match-winning advantage with the bat, but Shamar Joseph (2-26) removed openers Sam Konstas and Usman Khawaja before Alzarri Joseph (3-19) ripped through the middle order.
Number three Green remained undefeated, however, to prevent a complete Australia collapse and he will resume his resolute innings alongside his captain Cummins at the start of day three.
