Leicester Fainga’anuku and Codie Taylor punished disorganised England defence with tries for an early 12-0 lead but a barrelling Ollie Lawrence run and two George Ford drop goals cut the deficit to a point at halftime.
Tries by Sam Underhill and Fraser Dingwall early in the second half put England 25-12 ahead but Will Jordan struck back for New Zealand to make it 25-19 with 25 minutes remaining and ramp up the tension.
It was England's first victory over the All Blacks since the 2019 World Cup semi-final - they lost all three games last year - and first at Twickenham since 2012.
It was fully deserved too as they attacked with real venom and often left the vaunted All Blacks looking clueless with the ball.
England lost to New Zealand three times last year by a combined total of 10 points, failing to score a point after the hour in all of them. On Saturday, however, with a far more cohesive team, they kept the foot on the pedal to finish totally on top.
Ford had been preferred to Fin Smith at flyhalf for his vast experience and he showed every inch of it in a polished and composed display that earned a deserved man of the match award.
"I loved it, it was a challenging game," he said. "We went 12-0 down and came back well. We became a bit inaccurate so we had to work a few things out and the boys did that really well.
"We are big on trying to be calm and composed and give the right messages to the lads."
Fainga'anuku gives New Zealand fast start
After England had failed to turn early forays into points, New Zealand made them pay with their first opportunity as Fainga’anuku, the man with Twickenham as one of his middle names, squeezed through for the opening try after 15 minutes.
Two minutes later England’s defensive communication went horribly wrong to leave hooker Taylor in acres of space on the same left wing, and Beauden Barrett's conversion made it 12-0.
England’s pressure and willingness to try things paid off when Lawrence showed fantastic strength and aggression to blast through two tackles and put the hosts on the board.
Ford missed the conversion but ample amends with two crisply-struck drop goals to make it 12-11 at halftime.
Two minutes after the restart Taylor was yellow-carded for a pointless hand in the tackle and from the resulting penalty England sent hard runners at the line, culminating in Underhill charging over.
England then overcooked their next attack and gave away a series of penalties as New Zealand finally built pressure to send Jordan over for their first points for 47 minutes.
Ford slots over crucial late penalty
With six points in it the tension was palpable, but a lineout offence handed England a penalty six minutes from time 30 metres out wide on the right. Last year, Ford hit the post from a similar distance as England lost by two points, but this time he made the game safe.
The crowd roared in delight when the irrepressible Henry Pollock dribbled a loose ball into the corner for Roebuck to scoop up and score.
"To get the confidence to beat sides like this and hopefully go four from four in the autumn internationals is massive," said scrumhalf Alex Mitchell. "We're building and we've got so much in us still."
England will look to complete a sweep of their Autumn games and make it 11 wins in a row against Argentina next Sunday, while New Zealand visit Wales with their Grand Slam hopes over.
"England sort of ground their way back into the game. George Ford had a blinder and just accumulated points and they took their opportunities," said their captain Scott Barrett.
"At times we might have overplayed. England did pretty well at putting it in the air and chasing well. "We created enough, we just didn't get enough points."