Australia went to the tea break on Sunday at 8-187, still requiring 29 at the Gabba to deny the tourists their first Test win in Australia in 27 years.
In his second Test since replacing the retired David Warner as opener, Smith (76 not out) remained assured among the chaos.
He passed 50 to record his highest score of the summer, punching off the back foot and pulling with authority despite pressure mounting as wickets fell around him.
Chasing 216 to seal a 2-0 series victory, the hosts were cruising at 2-113 in the first session of the fourth day in front of a sparse crowd.
Enter Joseph (6-60 off 10), who in just his second Test bounced back from a nasty Mitchell Starc toe-crusher that forced him to retire hurt late on Saturday night to bowl a spell for the ages.
The 24-year-old reached speeds of close to 150kmh and found life with the old pink ball to bowl Cameron Green (42) and break a 71-run stand before crashing through Travis Head's defence with a yorker from his next delivery.
Two in two! Shamar Joseph making an impact in Brisbane! — cricket.com.au (@cricketcomau) #PlayOfTheDay | @nrmainsurance | #AUSvWI pic.twitter.com/FLMyIknCo7January 28, 2024
It was Head's second first-ball duck of the Test and incredibly his third in a row in Gabba Tests, the left-hander becoming the seventh Australian to register a king pair.
Smith survived the hat-trick ball but Mitchell Marsh (10) didn't last much longer, Alex Carey (two) Joseph's fourth in a manic six-over spell that also cost 45 runs.
He wasn't done though, Starc (21 off 14) Joseph's fifth scalp inside seven overs to end a brief cameo from the Australian quick that featured four boundaries.
Australia still needed 45 to win when Cummins, who blazed an unbeaten career-best 64 in the first innings before declaring 22 short of West Indies' first-innings 311, strode to the crease 10 minutes before the tea break.
The tourists chose to take another 15 minutes though when Cummins (two) edged Joseph to wicketkeeper Josh Da Silva and exposed No.10 Nathan Lyon.
Smith didn't shelter Lyon from the strike though, the pair navigating that passage to inch them closer to a 5-0 summer sweep.
9°C