The South Americans have drawn with Italy and defeated Slovakia in their two group stage matches and are now expected to qualify for the Round of 16.
But Santa Cruz wants more than that and believes the side can make it to at least the final eight for the first time ever.
"Paraguay has never been in the quarter-finals. That is our goal," he said.
"Maybe it is time to change history. We are very proud of our result today (2-0 over Slovakia). It will give us a lot of confidence.
"It is very important we win our last group game against New Zealand.
"There is a possibility we could top the table and we have to go for it. If we keep working hard who knows how far we can go.
"The atmosphere is fantastic not just in our games but in training," he added.
"We have been playing together for years and we have players who are mentally strong.
"Our strength is the spirit in the camp, the hard work we put into every game."