Woakes scored 19 from 15 balls to guide the critical last stage of England's run chase as it replied with 158-9 to Australia's total of 157-4.
The win gave England a 1-0 lead in the two-match series, the second of which is at Melbourne on Friday.
Australia allrounder Shane Watson almost single-handedly steered his side to victory, scoring 59 from 31 balls - the highest score of the match – and taking 4-15 to hamper England's chase.
Eoin Morgan topscored for England with 43 and Woakes showed composure beyond his years to steer his team home as it chased four runs from the last over.
England seemed headed for a comfortable win as it reached 50 inside five overs in front of 32,000 fans, and on a ground on which it achieved one of its three wins in the recent five-test Ashes series.
But Watson, already Australia's batting hero, almost turned the match when he struck with successive balls to remove the dangerman Morgan and Michael Yardy, prompting England's slump from 130-5 to 130-7 inside the 16th over.
England finally came to the last two overs needing 15 runs to win and with only two wickets in hand. Woakes and Graeme Swann combined to take 11 runs from the 19th over, bowled by Brett Lee, and to leave England needing four from the last six balls with two wickets in hand.
The last over fell to Watson and he immediately ramped up the tension on both sides when he dismissed Swann for 6 with the first ball. The action left England's final pair at the crease, the redoubtable Woakes at the non-striker's end and the last man in, Ajmal Shahzad.
Shahzad struggled for three deliveries to work the single that gave Woakes the strike and the chance to decide the match. Unfazed by the drama around him, Woakes worked the penultimate ball for two to tie the scores then crashed the final ball through wide mid-on for the winning run.
Earlier, Watson made the largest contribution in Australia's bid to set England an unreachable target. He hit six fours and three sixes in his 31-ball innings of 59, taking 26 runs from a single over of Swann's offspin.
