The two-time champions, who have never failed to reach the World Cup semi-finals, extended their record winning run to 29 consecutive victories and will face Canada in the last four clash at Eden Park next Saturday.
Hosts New Zealand, the last team to defeat the English back in 2019, will take on France in the other semi-final.
"We've spoken all week about trying to get a good start and we're used to these conditions back out in England, and it showed," said England captain Sarah Hunter.
"We can adapt and still play rugby. I'm really proud of the way the girls attacked this game. It's easy when you see it pouring down to shy away but the whole group fronted up.
"We're in knockout rugby and that's the sort of performance you want."
Hunter celebrated becoming the most capped woman in test rugby by giving her side the lead eight minutes into her 138th appearance as a huge shove took the English over the line.
A yellow card for Australia's captain Shannon Parry 20 minutes later was punished by the English, with Parker capitalising on the strength of her pack to score a pair of tries while the opposition was short-handed as Simon Middleton's side opened up a 19-0 lead.
That advantage was reduced late in the half when Emily Chancellor completed a flowing move from right to left to score an unconverted try for Australia, but Abbie Ward's pick-and-go score three minutes into the second half restored England's 19-point lead.
Packer fed Amy Cokayne to further add to her side's tally and Alex Matthews bulldozed over the line 15 minutes from time before Packer completed her hat-trick with seconds left in the game.
"We love to win but that wasn't the case today," Australia's Sharni Williams.
"We did absorb a lot of pressure, we knew England were going to come our firing and that's what they did. That first half we should be proud of our efforts, but obviously we couldn't hold on for that 80 minutes."
In the other quarter final, Canada ran in four tries to beat neighbours the United States 32-11 in an arm-wrestle of a contest.
McKinley Hunt, Karen Paquin, Paige Farries and Alex Tessier all crossed to send Canada to a fifth World Cup semi-final with a deserved victory in wet conditions at Waitakere Stadium.
"I'm so proud of our group," said Canada skipper Sophie de Goede. "(We have) so much belief. All our fans and our families all believe in us and now it's just time to go out and perform."
The United States, inaugural World Cup winners in 1991, gave everything they had but were unable to kick on after rattling their opponents with a bright start.
They took the lead in the ninth minute when hooker JoJo Kitlinski forced her way over the line from close range, but Canada quickly hit back with two tries in four minutes.
Lock Hunt got across the line in similar style but Paquin's effort in the 20th minute was from open play, with the flanker finding Elissa Alarie on the right wing and winning a foot race to the fullback's chip ahead.
American prop Hope Rogers had a try disallowed for a knock on in the 29th minute, but her team mate Alev Kelter cut the deficit to 12-8 with a penalty just before halftime.
Paquin helped give Canada a flying start to the second half when she stole the restart, and left winger Farries carved her way through the American defence for her team's third try.
Kelter again cut the lead with three points from the kicking tee but was almost immediately shown a yellow card for a dangerous tackle, allowing de Goede to extend the lead to 22-11 with the resulting penalty.
Canada's extra woman proved helpful in the 57th minute when they camped on the U.S. line and flyhalf Tessier was able to find space outside the defence to cross for the fourth try.
The Americans continued to pound away at the Canadian defence, but de Goede kicked her second penalty to take the lead to an imposing 21 points with 15 minutes to go.
"We talked last night about how, win or lose, we would walk off this pitch proud," said an emotional U.S. captain Kate Zackary. "I couldn't be more proud of them."
