With the two sides level at 1-1 after 90 minutes, Mikel Merino struck the decisive blow for Spain in the 119th minute with a brilliant header to send his side through to face Portugal or France, who were playing later on Friday.
"I'm proud of this group of players, such a committed group of players, who are an example for our country, for our society, and for any sportsmen and women who want to be competitive and want to improve every day in order to try and be a bit better every day," a delighted De la Fuente told reporters.
"Our team has heart and I know them, they have ambition and they are eager to be better every day. They never tire of improving and I am proud to coach them," he added.
The clash was littered with yellow cards and punctuated by an injury to Spain midfielder Pedri and a late sending-off for his team mate Dani Carvajal, but the coach said that he did not see it as a particularly dirty contest.
"I played in the 80s - if you want to watch a video of the 80s and see how football was played back then, you know I don't get scared. What do you want? I have a friend who says, 'what do you want, to get kissed?" De la Fuente said.
"This is football, I am not afraid of these things - the game is played to the limit, I am not complaining about the toughness of the opposing team, I rather appreciate what we as a team have done."
Spain will now focus on recovering for Tuesday's semi-final.
"We will see how far we can get, but I'm really proud of the example that this team is giving out in terms of companionship and values and so on, and this is what I value the most, apart from the quality of my players," De la Fuente added.
"We are happy, but we know that tomorrow is a different day, and tomorrow it's another game already. So yes, we are happy but the euphoria is very much under control," he added.