Chasing 169 for victory, Sri Lanka were dismissed for 87 as Australia won with more than three overs to spare at the Kensington Oval.
But Australia themselves collapsed to 30 for four and 67 for five before White, who made an unbeaten 85, and Michael Hussey (39 not out) shared an unbroken stand of 101 that took Australia to 168 for five.
Sri Lanka opener Mahela Jayawardene's previous scores this tournament were 81, 101 and 98 not out.
But the dismissal of the tournament's leading run scorer by Dirk Nannes, the tournament's leading wicket-taker, for just nine sparked a dramatic collapse.
Tillakaratne Dilshan (20) and Dinesh Chandimal (19) were the only Sri Lankan batsmen to make double figure scores.
Mitchell Johnson, like Nannes a left-arm fast bowler, who ended the match by bowling Ajantha Mendis for one, was the most successful Australian bowler with three wickets for 15 runs from 3.
2 overs.
"We are doing well, so far so good," said Australia captain Michael Clarke.
"Ther