Samoa have a points differential of plus-132 after beating France 78-4 and Papua New Guinea 64-6, leaving England with a steep climb towards the World Cup semi-finals.
England's 82-0 win over France in Perth lifted their differential to plus-64, but they must beat Papua New Guinea in Wollongong on Friday and hope Tonga defeat Samoa in Sydney two days later.
If both results go England's way, qualification will depend on points differential. Samoa remain in a strong position even if they lose to Tonga.
England, ranked third in the world, had needed a victory by around 80 points to keep their hopes alive after losing their opener 30-12 to Tonga.
They scored 14 tries, with Herbie Farnworth claiming a first-half hat-trick and Newcastle winger Dominic Young also crossing three times.
Young's opening try came after France had held England scoreless for 22 minutes. England then scored six tries in 18 minutes to lead 36-0 at halftime.
"What we've done today is enormous; given the context, I think it's just enormous," England coach Brian McDermott said.
"Knowing that we had to win big, we all had some candid conversations this week about what that actually means and what it would look like.
"Whatever the results are next week, we want to finish strong against PNG.
"If we go home next Sunday then that'll be a big disappointment, but I don't know what else we could've done."