The tourists beat England by 251 runs to win the first game of the five-match series.
England, set an imposing target of 398 to win, collapsed to 97-7 in the post-lunch session on the fifth and final day before being bowled out for 146.
Off-spinner Nathan Lyon took 6-49 and fast bowler Pat Cummins 4-32 as Australia enjoyed their first win in all formats at Edgbaston since 2001 -- the last time they won an Ashes series in England.
The match was a personal triumph for Australia's Steve Smith, who in his first Test since a 12-month ban for his role in a ball-tampering scandal, scored 144 and 142.
Smith was the hero for Australia as they battled back from 122-8 in the first innings.
"I thought we were certainly up against it at that stage but we had the best player in the world at the crease," said Paine.
"Steve showed his class in both innings. We're running out of things to say about Smithy. He's probably the best Test batsman we've ever seen. We're lucky to witness it.
"Nathan bowled superbly but our three quicks took a lot of pressure off him at the other end as well. The seamers were sensational.
"We thought we could improve on our first-innings performance with the ball. All those guys are world-class performers. This morning all four of them were sensational.
"We know every Test we play in England, the crowd will be against us. I thought the guys handled themselves really well and we've come out on top."