Gerrard provided the assist as the returning Wayne Rooney headed home from point-blank range for the winner just after half-time and he feels an unheralded squad can continue to improve.
England now face Italy in the quarter-final on Sunday.
He told ITV1: "It was a difficult group and the idea was to try and win it. No-one believed in us at the start but we're gaining momentum at the right time.
"I don't think tonight's [Tuesday] performance was fantastic, but we stuck together, got the job done and won 1-0 against a good team.
"We've been criticised in the past maybe for not turning up, under-performing and we can take that, we're man enough. But you saw a reaction.
"When you get criticised it's not nice and you have to react by playing better and I think that's what we did."
Gerrard brushed aside suggestions England would be overjoyed at playing Italy instead of world and European champions Spain, insisting the team were ready for whoever they may face in the remainder of the competition.
"Italy are next, they've got match-winners and we have to give them respect," he said.
"And if we do well in the next game we might play Spain again in the competition."
Gerrard admitted England had luck on their side when Ukraine striker Marko Devic's shot appeared to cross the line before being cleared by John Terry.
But he said: "To be successful in these tournaments, because of the standard of teams involved, you need that bit of luck going with you.
"Two years ago we didn't get that luck with Frank Lampard's goal, a big turning point in that World Cup game against Germany, and we ended up packing our bags and going home.
"Today [Tuesday] the luck turned. But it just shows that, if you keep working hard and keep fighting, you earn that luck. We deserved it today [Tuesday]."
Goalkeeper Joe Hart was beaten once in the match, when Marko Devic's shot apparently crossed the line before being hooked clear by John Terry.
Hart told Radio Five Live: "I haven't seen it, but the most important thing is we won the game 1-0."
Manager Roy Hodgson went further, admitting his side had a slice of fortune.
"They (the players) have been really good from the start. If someone said we'd win the group I don't think they'd have many takers.
"Most people didn't think we'd get out of the group. We've won it on points instead of goal difference as well.
"This was an away game with a capital 'A': there were 60,000 people in and only 4,000 brave England fans. When we needed a bit of luck, when the ball may have crossed the line, we got a bit of luck and went on to win the match."
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Hodgson, who had just two matches in charge before the start of the tournament, stressed his team were a work in progress.
He was, though, pleased that Rooney made a goalscoring return from a two-match suspension.
"This is a fledgling team as far as I'm concerned," he told ITV1.
"But we're working at our game and showing composure on the ball and in attacking areas is always the most difficult part.
"That's where you need your Wayne Rooneys. This game will be good for him.
"He played well, worked hard and could have had several other goals. On the counter we were dangerous enough but we could have been more dangerous."
Rooney was visibly buoyed by his return to the side.
"Obviously I was delighted with the win and to score," said the Manchester United man.
"I haven't scored at a major tournament for a while (since 2004) so it was nice to get a goal which won us the game.
"We stuck in well at times against a difficult crowd and overall we deserved the win."
Asked how he felt about Italy in the next round, Rooney was more eager to stress his own side's qualities.
"I said before it doesn't matter (who the quarter-final opponents are). The main thing was to get through the group and to finish top," he added.
"We know our qualities and what we're capable of. We've worked hard, done a lot of tactical play in training.
"We're difficult to beat and I don't think any team will fancy playing us."
Rooney returned from suspension to decide the outcome with the only goal early in the second period - his first in tournament football for eight years spanning 10 games.
But the supplier was Gerrard who turned in another man of the match display to help set up a last-eight meeting with Italy.
Rooney said: "Steven's delivery has been exceptional in this tournament.
"He's a world-class player and I thought tonight he was really brilliant.
"He drove us on at times. We're lucky to have a player like that in the team."
Rooney was delighted to settle the outcome after being banned for the opening two games against France and Sweden.
He said: "It was great to get three points and its always a great feeling to score.
"My overall game could have been a bit better but it was my first game for a while.
"The one thing I was delighted with was that I was always putting myself in goalscoring opportunities. I could have done better with a couple more but I got the goal and its three points."
Rooney celebrated the goal in bizarre fashion by pretending to spray his hair.
He explained: "I used Andy Carroll's hair product and before the game he asked me to do it if I scored - and so I did it."
England have avoided a meeting with defending champions Spain by topping their group but Gerrard has warned England that Italy will provide a stern test.
He said: "People say we've avoided Spain but I don't want to underestimate the Italians.
"Italy isn't going to be an easy game. Spain are a fantastic team, sure, but our plan was to finish top of the group.
"Now we're going to concentrate on the next match, which is Italy."