Cooper scored two tries, while captain Rocky Elsom also crossed for the home side, who were prepared to play an expansive game and defended well but were demolished in the scrum.
The Australian front row of debutant loosehead Ben Daley, hooker Saia Faingaa and tighthead Salesi Ma'afu, both of whom were playing just their second Tests, was constantly under pressure, with referee Nigel Owens awarding two penalty tries to England for repeated collapsing.
The Wallabies' up-tempo game and several linebreaks through a porous England defensive line, however, proved the difference.
"There was some great defence there," Elsom said. "The fact they didn't cross our line (in general play] was a big plus."
Australia's two first-half tries were the result of sweeping movements with Elsom finishing off after a midfield break, while Cooper took a clever inside pass from scrumhalf Luke Burgess. Both were converted by fullback James O'Connor to give them a 14-0 lead at the break.
Toby Flood opened England's account with an early second-half penalty before their scrum pressure paid off and, after a series of collapses by the Wallabies, England were awarded a penalty try, which Flood converted to bring them to within four points.
The Wallabies, however, struck straight back with Digby Loane feeding Cooper for his second try.
England then spent the best part of 10 minutes camped in Australia's 22, slamming into scrum after scrum, with Owens sinbinning Ma'afu then awarding his second penalty try after another collapse by the Wallabies.
The Wallabies, however, saw O'Connor slot a penalty before Cooper took over the goalkicking and put the match beyond doubt.
"We gave them too many opportunities," England captain Lewis Moody said. "We had some dominance (in the scrum] and it paid off but we have to work on some things for next week."
The second Test is in Sydney next Saturday.