The Three Lions were knocked out of the tournament in the quarter final stage on Sunday after going down 4-2 on penalties to Italy.
But Bernstein has praised both the players and new manager Roy Hodgson for making an immediate impact on the squad on and off the pitch.
Bernstein told BBC Sport: “That spirit started to grow again, and people started to talk about winning the competition and so on, so we've definitely done that.
"It's disappointing to lose a quarter-final again in the way we have, there's no question of that.
"It's not what we're trying to do, so the glass is half full, let's put it that way, but there's a way to go."
Despite being knocked out the tournament in Poland and Ukraine, Bernstein believes that England now have something to build on.
"I think Roy has made a real impact very, very quickly, just the sort of thing we thought he'd do. His organisational skills are very clear," he appraised.
"He was dealt not the easiest hand at such an early stage but I think he's played it very well, and I think there are positives and I think we have a good base to move forward from.
"The qualities we thought he'd bring, he clearly is bringing.
"You can ask the players and people around the squad, there's been a great feel, and I think, compared to previous competitions, we're a long way ahead of that, in performance as well, not just in spirit.
"We all know there are certain shortcomings in the way we play our game and we need to move forward in that, but that's not a lesson we've learnt today - we knew that before."
Bernstein also believes that the opening of the national football centre will help England become a better team and achieve more in future tournaments as the focus shifts to improving young players' technique.
“The opening of the national football centre this year is a terribly important thing going forward," he explained.
"In the youth development scheme the emphasis now is on small-sided football, skill-based football, passing the ball, less emphasis on physicality, all those things which are our agenda already.
"This is not coming from last night [Sunday], we've worked on this for years, but I think these things are coming to fruition now, so I do expect to see improvements - but it won't happen overnight."