Wayne Bennett's side were leading 20-0 with seven minutes remaining after tries from Jermaine McGillvary, Gareth Widdop and John Bateman.
But Tevita Pangai Junior, Siliva Havili and Tuimoala Lolohea crashed over at a sold-out Mount Smart Stadium.
Referee Matt Cecchin ruled that the prop had earlier lost the ball - and to the ire of the Tongans did not refer it to the video referee.
It was an absolutely pulsating, nerve-shredding end to a match played in front of a passionate crowd who heavily favoured Tonga.
England will face the Kangaroos, who thrashed Fiji 54-6 in the other semi-final, in the final in Brisbane next Saturday, 2 December - a match you can watch live on BBC One.
A breathtaking seven minutes of sport
England's last appearance in the final came 22 years ago with Denis Betts, now among the coaching staff, captaining the side in a 16-8 defeat by Australia at Wembley.
Every tournament since has ended in a semi-final defeat, each one by New Zealand, but it looked with seven minutes to go as though England were cruising into next Saturday's showpiece.
England's semi-final record since 1995
1995 - England 25-10 Wales
2000 - England 6-49 New Zealand
2008 - England 22-32 New Zealand
2013 - England 18-20 New Zealand
At 20-0 down, Tonga had other ideas.
First, after 73 minutes, Junior edged his way over the England line to cross for a try that was eventually confirmed by the video referee - but it looked like nothing more than a well-deserved consolation.
But the nerves began to kick in for Bennett's side as Tonga kept the ball alive before Havili broke through several would-be tacklers to make the score 20-12.
Two minutes later and England's advantage was reduced to two points, Lolohea again breaching England's line to put Tonga within one score of their first World Cup final.
England were in disarray and the crowd were absolutely roaring on Tonga.
Tonga so close to stunning England in semi-final
McGillvary, one of England's best performers at this World Cup, stole a key intercept to win the ball back with England just needing to see out the final 90 seconds, but the winger lost the ball in a tackle to give Tonga one last attack.
And when Fifita found a route through it looked like England's tournament was yet again going to end in semi-final heartbreak.
Elliott Whitehead's attempted tackle forced the prop to drop the ball but, with Tonga thinking Whitehead had ripped it in an attempted ball steal, Fifita re-gathered and touched down over the line.
Australian referee Cecchin deemed it as a loose carry from the Tonga prop, opting not to consult the video referee, and that was enough for England to hold on to reach a first final in 22 years.
Widdop inspires England in Auckland
Gareth Widdop
Gareth Widdop had a hand in all three of England's tries
For 70 minutes, England's scrambling defence had stood up to the Tongan threat while the forwards set the platform for Widdop to provide the spark from full-back.
England began their World Cup campaign with Widdop in the halves, but head coach Bennett used the St George Illawarra Dragons man the number one shirt in the final group game against France.
Widdop has kept that spot since and, while also allowing Kevin Brown to flourish at six, proved the inspiration against Tonga with a hand in all three of England's tries.
It was his superb break that set the field position for England to score their first, joining the line to send McGillvary in at the corner.
The full-back doubled England's advantage shortly after with a try of his own, gathering Whitehead's pass to roll over from close range for a score that was awarded by the video referee.
Widdop then edged England further ahead with a penalty after the break, before teeing up Bateman with a well-timed pass for the centre to crash in for England's third.
He played with a positional nous and cunning that England will need if they are to threaten an unbeaten Kangaroos team next weekend.
A tough test in 'Little Tonga'
Tonga
An estimated 80,000 people of Tongan descent live in Auckland
Papua New Guinea's matches in Port Moresby were all sold out but very few other matches have been at the World Cup and there were oceans of empty seats at Friday's other semi-final in Brisbane.
But England were met by a sea of red shirts and flags as the Tongan support poured into a sold-out Mount Smart Stadium which, at 30,003, witnessed its biggest crowd since home side Auckland Warriors joined the Australian Rugby League in 1995.
There is an estimated 80,000 people of Tongan descent in Auckland, leading the city to be dubbed 'Little Tonga' and it almost made the semi-final feel like a home game for the Pacific Islanders.
"I've played in some massive games and hostile atmospheres, but nothing like this before," said England prop James Graham.
Tonga's desire and belief they could get back into the semi-final was characterised by a rousing chorus of hymns from the red and white crowd, while they shared in the players' tears at the final whistle.
Tonga became the first Tier Two nation to beat a Tier One side for more than 20 years when they beat the Kiwis in the group stage, coming after the likes of Jason Taumalolo, Manu Vatuvei and Fifita turned down New Zealand and Australia to play for the countries from which their families descend.
England did well to stem the initial tide of enthusiasm, looking solid defensively and enjoying a 100% success rate in their opening 10 sets.
Mount Smart Stadium was sold out for the match
Passionate, committed and absolutely amazing as they sung as one - the pro-Tonga crowd were absolutely amazing
Bennett's side were let off when the Pacific Islanders shifted a flowing move to the left that winger Daniel Tupou fumbled with a clear run of the line in front of him.
But with Brown and Luke Gale in the halves, England's game management was more controlled and considered - until a frantic final few minutes - and it was for large parts their most convincing performance of the tournament against undoubtedly their toughest test so far.
"For very long periods we were on top of that game but a little ill disciplined when we needed to be better," said Tonga coach Kristian Woolf.
"For whatever reason the sorts of opportunities we have taken in past games we weren't able to take."
Australia await
Australia 54-6 Fiji highlights
So Australia, for the 14th successive time, await in the final in Brisbane next Saturday.
The clinical Kangaroos brushed aside a Fiji side in the semi-finals that had shocked co-hosts New Zealand in the last eight.
Valentine Holmes crossed for a record six tries, his 12th of the competition, and has already surpassed Wendell Sailor's record of 10 scores with one match left to play.
But Australia, unbeaten under coach Mal Meninga, are a threat right across the park with a settled back line, powerful forward pack and an in-form halves partnership led by Cooper Cronk.
England will have to call on all that scrambling defence and ingenuity in attack if they are to shock the holders on their own turf.
Tonga: Hopoate, Tupou, Jennings, Hurrell, Fusitu'a, Lolohea, Hingano, Fifita, Havili, Taukeiaho, Ma'u, Manu, Taumalolo.
Replacements: Katoa, Terepo, Pangai Jnr, Murdoch-Masila.
England: Widdop, McGillvary, Watkins, Bateman, Hall, Brown, Gale, Hill, Hodgson, Graham, S. Burgess, Whitehead, O'Loughlin.
Replacements: Walmsley, T. Burgess, Currie, Roby. (BBC)
