Brian McDermott's world number three side looked set for a big score when leading 22-0 after 20 minutes at Wollongong, south of Sydney, but they fell apart as the game wore on.
They now await the outcome of a crunch clash between Tonga and Samoa in Sydney on Sunday, which will determine whether they stay alive or make their first group-stage exit in World Cup history. If unbeaten Samoa win, England head home.
Even a victory for Tonga may not be enough, with England's fate resting on points difference. They need Tonga to win by at least 46 points.
Only two teams from the six nations in Groups B and C progress.
Defending champions Australia have already booked their place in the last four, with New Zealand and Fiji clashing on Saturday for the right to join them from Group A.
"We were good the first 20 minutes, and then the wheels kind of fell off in too many areas," said England captain George Williams.
"And credit to PNG (Papua New Guinea), they didn't stop fighting.
"We're relying on Tonga to do a more than massive job. There's a chance but a very slim one."
England had a nervy start but settled into their groove and winger Matty Ashton dotted down in the corner to open their account after seven minutes.
They were laser-focused and doubled the score with a second converted try, this time from AJ Brimson on the counterattack, to lead 12-0.
PNG were desperately defending and barely getting a look in.
Despite numerous handling errors, an inevitable third try - from Dom Young - soon came, followed by another minutes later under the posts by Harry Smith.
But England lost their mojo and PNG prevented further carnage until conceding a penalty on the half-time buzzer to send England into the break leading 24-0.
Against the odds, PNG opened the second half with consecutive converted tries in a huge setback to England's semi-final hopes.
Dudley Dotoi got under a high kick and crossed, then Gairo Voro steamed through the defence for another to close the gap to 24-12.
PNG had all the momentum and were a constant threat before England finally bagged another try, 47 minutes after their last, courtesy of Morgan Knowles rolling over.
One more at the death from Mikey Lewis ensured a flattering scoreline.