Three tries in eight second-half minutes from Brad Barritt, Chris Ashton and Manu Tuilagi saw England secure their biggest-ever win against the All Blacks in an absorbing Test match.
"There were two teams capable of winning the World Cup out there," said New Zealand coach Steve Hansen. "No excuses, we got beaten by the better side. This is a good England side. You should get behind them and back them a bit. They have shown what they can do, and if you back them it will make them even more dangerous."
He added: "They thoroughly deserved their victory and should be proud of what they have achieved with magnificent football. They took the game to us from the get-go, full credit to them."
England, who had lost narrowly to Australia and South Africa in the past two weeks, took a 15-0 lead at half-time, courtesy of four penalties and a drop-goal by fly-half Owen Farrell.
And while New Zealand hit back with two tries from Julian Savea and Kieran Read to get within a point at 15-14, England regrouped in spectacular fashion to take the game away from the world champions, who gave some respectability to the scoreline with a late second try from Savea.
Lancaster said: "The performance was exactly what we had hoped for. I'm chuffed for the players. When the tough times came we stayed together. That team belief with a young England side definitely augurs well for the future.
"At 15-14 we came back with three tries of our own. We've tried to instil a no-fear mentality in the players and to make them have the courage of their convictions.
"To build a 15-point lead was very pleasing. The All Blacks came back strong but our energy and our fight for the shirt was outstanding. That's what got us across the line.
"Our defence was outstanding. We forced the All Blacks into errors and not many sides do that."