Darren Sammy won the toss and the tourists backed up their captain's decision with four spectacular catches in the field at the top of the Australia innings.
But, after Australia fell to 98 for six when Matthew Wade was out, Bailey (125 not out) and James Faulkner (39) resurrected the innings with a 100-run partnership.
Bailey brought up his ton off 105 balls with a six over wide long-on off the final ball of the 49th over and then bludgeoned 25 off Dwayne Bravo's final over, which included three sixes and a four.
Australia had made a steady start to their innings before Usman Khawaja (three) fell victim to a Kieran Powell full-length diving catch to his right off the bowling of Kemar Roach.
Aaron Finch (11) followed his opening partner shortly after when he was caught by Darren Bravo at point off a hard-hit cut shot, Jason Holder the bowler this time.
When Phillip Hughes (21) and Michael Clarke (16) fell within three balls of each other, Australia had slumped to 56 for four in the 13th over.
Wade and Bailey helped negotiate the first part of the middle overs and put on a 37-run partnership.
Wade was dismissed by Sunil Narine as he attempted a cut shot, but only succeeded in a top edge, which went high to Sammy's left at first slip and he stuck out a hand and clutched the chance.
And, as it did with Hughes and Clarke, one wicket brought two as Glenn Maxwell was out for a golden duck as he chopped on in an effort to shoulder arms.