A first-half goal from the recalled Jermain Defoe was sufficient to secure victory in Port Elizabeth as England finished second in Group C behind the United States.
And memories of the woeful showing against Algeria were erased as England created enough opportunities to have won by a larger margin.
Lampard said: "We are delighted. It was a game we had to win and we did that with a good performance.
"We could have scored more goals and made it more comfortable rather than having a nervy ending.
"But, in terms of the performance, we spoke in the week in terms of playing with a bit more passion and a bit more pace in our game.
"We certainly did that in the first half. That was the difference."
Lampard is confident England can make major progress in the tournament if they maintain that level of spirit and intensity.
The Chelsea midfielder added: "We play that way for our clubs. That is why people regard the Premier League as one of the best and fear it when we play at our best.
"If we play with those qualities, then we have got a great chance."