It was end-to-end action at a sold-out Parramatta Stadium as Tonga headed to the break with an 8-4 lead before withstanding an intense second-half barrage to seal a bruising win.
Their reward is a semi-final next weekend against New Zealand while Samoa will meet defending champions Australia, who beat them in the 2022 final.
World number three England had a slim chance of staying in the tournament ahead of the game, but needed Tonga to beat Samoa by at least 46 points.
Instead, Brian McDermott's team make their first group-stage exit in World Cup history.
"This is probably one of the best games I've been a part of," Tonga captain Jason Taumalolo said.
"We knew it was going to be a physical game, and it was going to be tough right from the start. We got ourselves a little lead, but, you know, Samoa being an electrifying team made it a contest right to the last minute."
The game began at a ferocious pace after both teams performed their ceremonial dances - the Sipi Tau and Siva Tau.
Tonga had the best of the early chances with two tries disallowed.
It took them until the 27th minute to finally breach the Samoa line when Sydney Roosters centre Rob Toia skipped through two tackles to dot down under the posts.
Isaiya Katoa converted and then added two more points from a penalty before Samoa hit back on the cusp of half-time, when Brian To'o collected an offload to send them to the break trailing 8-4.
Both sides bagged early tries in the second half, with Jojo Fifita over in the corner for Tonga before To'o dived across for his second.
Samoa sparked to life and, with Tonga tiring, Hamiso Tabuai-Fidow and then Ronald Volkman collected tries to give them an 18-14 lead with 10 minutes left.
But Tonga never gave up and Addin Fonua-Blake charged over with six minutes remaining, and Katoa kept his cool to nail a crunch conversion for a 20-18 advantage.
In an earlier dead-rubber group game, Lebanon pipped France 24-20 with Abbas Miski crossing in extra time for a thrilling golden point win.
