In a repeat of the 2023 final, England made a strong start with two goals in the first 15 minutes from James McAtee and Harvey Elliott.
Spain replied from the penalty spot before the break and pushed for an equaliser in the second half.
However, England held out and Elliot Anderson made sure of the win with an injury-time penalty.
England next face Netherlands in the last four on Wednesday evening and manager Carsley has his sights on lifting the trophy again.
“This can’t be our final, this can’t be the highlight of our competition,” he told Channel 4 after the match.
“There’s better nights to come and it’s good now that we have an extra day to recover. We are looking forward to the next game.”
England had failed to win their previous two matches at the tournament, drawing with Slovenia and losing to Germany.
But they were impressive from the start against Spain as they took control with two early goals and then held out defensively in the second half.
“Lost for words,” Liverpool’s Elliott told Channel 4.
“I said at the start of the game it was going to be very tough and they would play an incredible game against us, but the way the lads handled the pressure, conceding a goal and being determined to keep on going and fighting, we deserve it so much.”
Asked how much England had improved on their group displays, Elliott added: “There was a major difference – from start to finish we were compact, we were a team, we fought together, we didn’t give up, and most importantly we trusted in our own ability.
“We need to take a lot of things from this game into the next one.”
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