The South Americans beat Japan 5-3 on penalties on Tuesday to reach the last eight for the first time following three previous disappointments at the second round stage. But 28-year-old Santa Cruz said that success has made the team hungry for more and they will not be content with their showing if it ends here.
"This (record) is more for the people in Paraguay, we're football players, we're used to playing in teams that win every game," said Santa Cruz, who endured an injury-hit season with Manchester City last term. Even if we're very pleased and happy that we qualified (for the quarters), we didn't expect to go to penalties (against Japan), we expected to be far better than we have been playing. We have not reached our ceiling yet, we want to keep improving, play better and keep progressing.
"We can't feel what they are feeling in Paraguay, it's a football thing," he added. "The thing is that now that you are there, you think about keep on moving forward. We haven't been having a party because we set our goals very high and now that we're here we want go out and do whatever it takes. We want to plan our game and see if we can put it into practice, we have full confidence that if we do a proper game, we can win."
Having enjoyed the lion's share of possession against Slovakia, New Zealand and Japan in their last three games, Santa Cruz - who is playing in his third and what he says is likely to be his last FIFA World Cup - knows full well that it will be a different story against the talented Spaniards.
"Spain are a very compact team which has been playing together for a long time, everyone knows they're the favourites," he said. "It's going to be a tough game, Spain are very famous for managing the ball very well and the ball possession won't be on our side as it was the last two games.
"But it will be one of those games that we're very used to playing in South America when we play against Brazil or Argentina so it's not a type of game that we're not used to playing," he continued. "We're full of confidence that we can play a great game. We'll play the same game we usually play and try to get the ball very early, that's something that won't change."
However, Santa Cruz said he doesn't want his team to play like they did in their opening game of the tournament against then reigning champions Italy, where they hardly got over the half-way line in a 1-1 draw.
"That was probably our worst game that we've played so far, I don't think we will use that game as an example of what we want to achieve against Spain," he added. "We want to be far better in the possession of the ball and to not remain so deep, even if we defended fantastically it's not the game we want to play."
Paraguay's strikers have been much criticised for not having scored in their four games in the tournament but Santa Cruz insisted that is not indicative of how they are playing. "Our team is set up so we can press very early and high up the pitch so because of that we (strikers) do a lot of work defensively," he said. "Even if we play three strikers we know the guys playing either side will make a lot of sacrifices and get back to win back the ball early so we can get pressure on other teams.
"It's been working fantastically so far. The chances are there so if we haven't scored it's not that we haven't continued on from qualification where we were scoring for fun in the national team for four years. Now we're not doing it but it's just a matter of putting it in the net and now is the right moment to do so."