The tournament favourites had to claw back from 10-0 down midway through the first half as England cast aside a troubled build-up.
Playmaker Johnathan Thurston was instrumental as the Kangaroos ultimately had too much class in the tournament-opener match played under the roof at Millennium Stadium.
Australia coach Tim Sheens paid tribute to England, who could lose star forward Sam Burgess to suspension for a high tackle that forced Sam Thaiday off in the second half.
But Kangaroos captain Cameron Smith said the Australia needed to lift considerably in their bid to make amends for the shock loss to New Zealand in the 2008 final.
"It wasn't our best performance by a long shot," Smith said.
"We were pretty ordinary to be blunt in that first 20 minutes but I was really happy with the boys' composure.
"We were under a lot of pressure there but we found a way to come back."
A burst of three tries in 13 minutes allowed Australia to take an unlikely 18-10 lead into halftime after an error-riddled performance opened the door for an English ambush.
"I think a lot of it was to do with a lack of game time we've had," Smith said of the poor start.
"Not too many of us have played in recent weeks.
"In saying that, we are the Australian side we expect a lot more of ourselves ... we've got plenty to work on."
The entertaining-but-scrappy win means Australia are likely to avoid the reigning-champion Kiwis until the final.
"But our goal now is to play really hard over the next two games (pool matches against Fiji and Ireland) to improve ourselves and not drop our intensity," Sheens said.
Man-of-the-match Thurston scored Australia's first try and had a hand in two others while Andrew Fifita, the only debutant in an experienced side, had a solid debut off the bench.
South Sydney twins George and Tom Burgess impressed on debut as they joined Sam to became the first trio of brothers to play for England.
Warriors-bound fullback Sam Tomkins was also among England's best.
Steve McNamara's side had entered the match on the back of a friendly loss to minnows Italy and the axing of forward Gareth Hock for breaking a team alcohol ban.
They were also without dropped Canterbury enforcer James Graham but the dramatic build-up didn't show.
England's ambition was clear early as they twice knocked back gift two-point penalties and their boldness was rewarded with tries to winger Ryan Hall and centre Leroy Cudjoe.
Australia slowly grew into the game and scored in the 27th minute, with Greg Inglis climbing above Tomkins to collect a Cooper Cronk bomb and finding Thurston in support.
Thurston put Greg Bird into a hole to give Australia the lead four minutes before Australia's players' player Billy Slater ran almost 60 metres to score on the stroke of halftime.
Josh Morris finished a sharp backline move to give Australia an ideal start to the second half and it looked like they could run away with it.
But England stayed in the fight, getting back to within a converted try when George Burgess crashed over in the 51st minute.
Thurston gave Australia an eight-point buffer with a 65th minute penalty and the North Queensland playmaker was again involved as Darius Boyd crossed with ten minutes to go.
Winger Josh Charnley capitalised on some sloppy play from Australia with a late try but England left it too late.
"We got a lot of positives from it, but there is a real sense of frustration that we didn't quite go on and finish the job off," McNamara said.
Burgess was reported for a high shot that left Thaiday dazed and in serious doubt to face Fiji in St Helens next Saturday.
Meanwhile, Wales coach Iestyn Harris offered no excuses after his men were outgunned 32-16 by Italy at the Millennium Stadium.
Wales led 16-14 early in the second period, but then conceded 18 unanswered points to a strong Italian side captained by former Australia fullback Anthony Minichiello and inspired by two-try centre Aidan Guerra.
Wales, boasting a totally home-grown squad, must now beat the USA and Cook Islands in their remaining two group games to have any realistic chance of reaching the quarter-finals.
"I thought the best side won, I think it's fair to say that," Harris said of the Azzurri, boasting Australians of Italian heritage.
"They had a bit more class in a few little key areas, but we've got a great spirit in the Welsh side. My players gave everything they got, and I won't say anything against them.
"I thought we had an opportunity when we put our noses in front, but we dropped a couple of balls when we had a bit of momentum. When you have got momentum in games you have got to take advantage, and we didn't do that."
Elliot Kear, Rhodri Lloyd and Ben Evans scored tries for Wales, while Lloyd White kicked two goals, but it was not to be for Harris' team as Italy powered clear during the second half.
Man-of-the-match Josh Mantellato scored a hat-trick of tries for Italy, Aiden Guerra a brace, and Chris Centrone and Mark Minichiello bagged one apiece in the cross-pool fixture.
Italy coach Carlo Napolitano paid tribute to his NRL-experienced side.
"Playing in our opening game of the World Cup, it is very important to get off to a good start. I am very proud of my players," he said .
"All we are thinking about now is next week and concentrating on Scotland. Then we will work towards Tonga.
"You can't get ahead of yourself. You have to take it one step at a time.
"It wasn't a perfect game of rugby league, but what is? We have got things we know we can improve on throughout the week, and that is what we will do. We will fix what we need to fix."