Owen took a wicket with his medium pace bowling before returning to smash 50 off 27 balls as Australia chased down a 190-run target with seven balls to spare.
West Indies, who lost the preceding test series 3-0, got off to a strong start after being put into bat at Sabina Park.
Captain Shai Hope (55), Roston Chase (60) and Shimron Hetmyer (38) laid the platform for a 200-plus score but the late flourish they expected never materialised.
Andre Russell, who will quit international cricket after the second T20 on Tuesday, made only eight.
Australia's Ben Dwarshuis (4-36) mowed down the middle and lower order as West Indies were forced to settle for a modest 189-8.
While Australia lost both their openers, captain Mitchell Marsh and Jake Fraser-McGurk, inside the powerplay they were propped up by a breezy 80-run stand by Cameron Green (51) and Owen.
Green belted five sixes in his 26-ball blitz, while Owen smacked six of them before holing out in the 17th over with Australia on the cusp of victory.
