Bailey put on 100 from 123 balls for the seventh wicket with James Faulkner (39) to rescue the hosts from a parlous 98 for six less than half way through their 50 overs.
The 30-year-old saved his best until the end of the innings, though, and after reaching his century with a six over long-on in the penultimate over, hammered three sixes and a four off the last six balls to finish unbeaten on 125.
The Caribbean islanders were highly embarrassed by their nine-wicket defeat in the opening clash of the five-match series last Friday and their fielding at least was much improved.
Four of the first five Australian batsmen to fall were caught while the fifth, skipper Michael Clarke, was clean bowled for 16 by Dwayne Bravo.
For pure reflexes, Darren Sammy's effort to dismiss Matthew Wade (16) was the pick of the bunch as the skipper snatched a rocket of a ball out of the air with one hand to leave Australia struggling at 93 for five.