Chasing 267, the visitors were rocked by in-form Starc who picked up two wickets in his first over.
Starc went on a rampage as he picked up his third in the eighth over to reduce the West Indies to 33 for three.
Opener Kieran Powell and Dwayne Bravo brought their side back into the chase with a 126-run stand for the fourth wicket. But Glenn Maxwell, the new million-dollar boy of the Indian Premier League, struck to give his side a vital breakthrough when he sent back Bravo.
