Gerrard, in his 95th appearance for his country, was named man of the match after an impressive midfield display including the cross for Wayne Rooney's 48th minute winner.
England topped Group D ahead of France while eliminating Ukraine, who miss out with Sweden.
"This was a very difficult group. Not many people had belief and confidence in the team but I always kept belief and confidence because I know we have very good players in the dressing room," he said.
England have stumbled in recent World Cups and European Championships despite high expectations from fans and media but Gerrard said that this side were living up to their potential.
"In previous tournaments we have underperformed and not delivered but we showed the signs of a good team - that when it is hard you don't give up and keep fighting," said the Liverpool midfielder.
The departure of Fabio Capello - leaving little time for new manager Roy Hodgson to mould a team - along with a number of injuries, led to a downgrading of hopes for England but Gerrard rejected the notion the team were "overachieving".
"I don't think we've overperformed, I think we are performing to the level we are capable of," he said.
"I've said before the reason we've gone home in other tournaments is that we didn't play the way we were capable of."
Hodgson said that given England were now in the last eight, it was time to move on from talk about expectations.
"It's more important to talk about the way the team has conducted itself in the tournament. It has been a really good tournament for us and we've really enjoyed it," he remarked.
"The fans have made a fantastic effort to support us, as have people back home because we get good vibrations from messages we are hearing and what we think people are thinking about the team...
"For us I don't think it is really a question anymore of expectations, we want to keep going and try to enjoy this tournament for as long as we can and play as well as we can and who knows where it will takes us?
"We are very pleased we have won the group, I don't think it was an easy group to win."