England were never in the contest after they suffered early losses on day five before a batting collapse on either side of lunch rendered them all out on 146 while chasing the mammoth target of 398.
Australia's win was headlined by Steve Smith's dream comeback to Test cricket, racking up centuries in both innings in his first game in the longest format since the expiry of his 12-month ban for his role in the Sandpapergate scandal.
Smith scored a crucial 144 in the first innings to rescue Australia after they had capitulated to a perilous 122-8 to lift his team to 284.
His valuable contribution of 142 in the second innings cemented Australia's dominant status in the game allowing skipper Paine to declare on 487-7.
Australia's emphatic win at Edgbaston is their first victory in any format on the venue in the last 18 years.
"You've got to keep a lid on it," said Paine, reminding his players to keep their emotions in check ahead of the next game.
"There's still four Tests to go and we aren't here to win the first Test at Edgbaston –- we're here to win the Ashes," he added.
"We've been really clear on that for some time. We’re obviously happy to win the first Test. It's a huge step in the right direction, but we’re certainly not satisfied with that."
Australia are still in search of their first Ashes series win on English since 2001 but Paine believed the current team under his leadership had the early advantage to rewrite history.
"We're over here to do something that a lot of teams from Australia have struggled to do," he added.
"And we realise that if we can do it will be spoken about for a hell of a long time, and that's what is driving us."
Former captain Smith, who was stripped of his leadership position in the aftermath of the ball-tampering controversy, was often observed to be calling the shots in the field with Paine taking the role of a deputy.
But Paine insisted he was keen on learning the tactical nous from the senior duo of Smith and David Warner to aid improvement in his captaincy.
"It certainly helps," said Paine.
"They've got things they've learnt through experience that other people don't have or don't know. We'd be foolish not to tap into that.
"They've still got a huge presence in our dressing room, there's no denying that."