Australia ended a rain-interrupted day three on 7-221, with Alex Carey (26no) and captain Pat Cummins (4no) still there when stumps were called due to bad light in Grenada on Saturday.
With the pressure mounting on their faltering top order, having slumped to 2-12 following the dismissals of openers Sam Konstas and Usman Khawaja late on day two, Smith (71) and Green (52) stepped up.
Green has struggled since returning from a serious back injury for last month's World Test Championship final defeat to South Africa.
Batting in the coveted No.3 spot, the 26-year-old had managed just 48 runs across his first five innings.
But Green looked assured at the crease until he chopped on to his stumps, a ball after reaching seventh Test half-century, leaving him furious.
Smith, who was dismissed cheaply for three in the first innings just 20 days after returning from a nasty finger dislocation, was chanceless as Australia hunt a series-clinching victory at St George's.
Smith was out lbw to Justin Greaves (2-22) when he appeared to be storming towards a 37th Test century.
Travis Head (39) helped maintain the impressive Australian run rate but fell to Shamar Joseph for the second time in the match.
Earlier, nightwatchman Nathan Lyon did well early on to support Green, eating up valuable overs as the pair sought to blunt the new ball.
Spin-king Lyon had batted an hour and 47 minutes, using up 33 balls, before departing to a catch from John Campbell at third slip off Alzarri Joseph.
His resistance made it easier for Green to settle in the first session and the West Australian looked determined to take on the responsibility of holding the innings together.
Pat Cummins' Australia side are looking to claim a series-clinching victory at the National Stadium in St George's, a venue in which they've never before played a Test.
The pitch is still tricky to bat on, ensuring any lead close to 300, combined with Australia's star-studded bowling attack, will make it difficult for the West Indies to chase down in the fourth innings.
