Openers David Warner and Cameron Bancroft put on an unbeaten 173-run partnership to help the hosts overhaul England’s target before lunch on the final day.
Warner, unbeaten on 87, and debutant Bancroft, who was 82 not out, broke an 87-year-old Test record for the all-time highest unbeaten opening partnership in a successful Test chase.
Australia needed on 56 runs to go one up in the series following a disappointing batting display by England on Sunday.
The visitors were dismissed for just 195 runs in their second innings after having conceded a 26-run lead in the first. England lost their last six wickets for 82 runs setting Australia an easy target in the remaining four sessions of play.
Unbeaten at Brisbane for 29 years
With this win, Australia stretched their unbeaten record at the Brisbane ground to 29 years since their last defeat the West Indies in 1988.
“We had to show some really good character throughout,” said Australian captain Steve Smith, whose crucial first-innings helped his team secure the important lead. "Obviously losing the toss, and the wicket was a lot slower than we anticipated it would be. We had to work really hard."
England captain Joe Root said the result did not tell the full story.
“For three days we were excellent. Bar Steve’s knock we were right in it," he said, referring to Smith's battling first innings century. “It’s frustrating. We have to move on quickly and make sure we don't make the same mistakes in Adelaide."
The second of the five-Test series will be held at Adelaide Oval starts on Saturday for the first-ever Ashes day-night Test.