Paraguay have reached four successive World Cups and have become recognised as the third football power in South America behind Brazil and Argentina.
Many games in the Paraguayan championship are played in front of only a few hundred people and infrastructure is poor, making it difficult explain how the country has done better than more populous, richer nations like Colombia, Peru and Chile.
"It's a miracle, it's something Paraguay has been blessed with and we are very proud, coming from such a small country, to be involved in such a great tournament," Santa Cruz told reporters after Saturday's 1-0 friendly win over North Korea.
"It's probably because we are so down-to-earth. The football is not the best, we don't have a lot of resources, we have don't infrastructure or stadiums like other countries have."
As if to prove Santa Cruz's point, his team mates stood around in the open air drinking herb tea known as Yerba Mate as they waited to board the bus after playing at the tiny stadium on the edge of Lake Geneva.
Santa Cruz said he felt confident and in good shape despite an injury-plagued season at Manchester City, where he played only 19 league games and scored three goals following his move from a much more successful spell at Blackburn Rovers.
"Fitness-wise I feel very good, and it's good for me to have 90 minutes behind me," he said after converting a late penalty to give his team victory.
"It's one of those things you never know," he said when asked whether his lack of action could turn out to be a benefit.
"It could be a good thing or it could be a terrible thing, I think our performance will speak for itself."
The 28-year-old is heading for his third World Cup and was hopeful that Paraguay, who face Italy, New Zealand and Slovakia in Group F, could reach the last eight for the first time.
"Paraguay have never been involved in the quarter-finals, that is what we want to achieve and see if we can go on from there," he said. "It would be a good World Cup if we could do that."
After facing a blanket North Korea defence in the friendly, Santa Cruz said he expected more of the same in South Africa.
"It will be useful for us, as we have to play teams like New Zealand and Slovakia, where probably we are the ones who are going to have more of the ball and will have to try to make the most of it.
"It's very tough when you get teams like that "
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