Chasing the home team's 9-205, Australia reached their target with three wickets and four balls to spare.
They looked to be cruising at 2-129 at the halfway stage, after taking 21 off the 10th over, but lost 3-5.
Green (55 not out off 35 balls) and the recalled Aaron Hardie (23 off 16) added 61, and the loss of two late wickets wasn't significant.
Green, who scored his third half-century of the series, Inglis (51 off 30) and Maxwell (47 off 18) produced the batting fireworks.
Maxwell, named man of the match, thrashed six sixes, including a trademark helicopter flick and a one-handed swipe over long-on.
The Australian catching was exceptional, with Maxwell going low and leaping high to take two excellent efforts at mid-off in the powerplay, and Mitchell Owen taking a diving snare at deep backward point.
Maxwell's excellent athleticism shone again late in the innings, when he took a high catch at long-on, and before falling over the boundary line he flicked the ball back to Green to complete the dismissal.
"I put a lot of emphasis on my fielding, I think it's one of the things I've always held a high regard for," Maxwell told broadcaster ESPN.
The West Indies innings was full of brief explosive batting cameos, with Sherfane Rutherford (31 off 15), Romario Shepherd (28 off 18), Rovman Powell (28 off 22) and Jason Holder (26 off 16) unable to push on.
Wickets fell at regular intervals, but the home side maintained a strong run rate, smashing 21 fours and 12 sixes.
Nathan Ellis (0-21 off four overs) and Hardie (2-24 off four) both bowled tidily on a small ground with short boundaries.
Xavier Bartlett finished with 2-39 off his four overs.
Paceman Sean Abbott (2-61 off 4) and legspinner Adam Zampa (3-54 off four) bore the brunt of the assault.
Captain Mitchell Marsh was given out lbw for a second-ball duck, though technology showed he erred in not reviewing the decision, because the ball pitched outside leg stump.
Inglis, who scored 78 not out in the second game, smashed the ball all around the ground.
He added a six to 10 fours, one of which was an audacious reverse pull.
Maxwell was largely a spectator in the first five overs, facing just three balls and scoring only one of Australia's first 49 runs off the bat, while Inglis tore into the West Indies bowlers.
Inglis's pyrotechnics came to an end one ball after the powerplay when he hit a full toss to deep backward square leg.
Maxwell was out first ball after the mid-innings break, swatting the ball to long-on, and Owen and Cooper Connolly fell cheaply to catches off left-arm quick Jediah Blades (3-29 off four).
West Indies dropped three catches and bowled 16 wides as they extended their record for the most unsuccessful T20I defences of scores of over 200 to seven times.
Captain Shai Hope rated their fielding across the seven tour matches, including three Tests, as "sub par".
Australia can compete an eight-match winning sweep of the tour by taking out the final T20I in St Kitts on Tuesday.
