Santa Cruz helped his side move in sight of the second phase with a 2-0 victory against Slovakia in a Group F match at the Free State Stadium.
Goals from Enrique Vera and Cristian Riveros either side of half-time gave them the victory they craved following the draw with reigning world champions Italy.
Santa Cruz, back in the team after a minor injury, said: "Paraguay has never been in the quarter-finals. That is our goal. Maybe it is time to change history. We are very proud of our result today. 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."
Santa Cruz believes the bond between Paraguay players is second to none.
He said: "We controlled the game and are happy to have achieved our goal of getting the points. Our strengths are as a team we are very compact. We are all proud to be part of the squad. The atmosphere is fantastic not just in our games but in training.
"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."
Paraguay coach Gerardo Martino was pleased with the way his side took the game to Slovakia.
He said: "We ran the match and our play in the first half was excellent. Then we had some moments of uncertainty but nothing really to be concerned about. Our objective now it to make the last 16 and we have a good possibility of doing that. What this victory will do is improve our spirit and make us mentally stronger."
Slovakia are in the finals for the first time and coach Vladimir Weiss refused to be downbeat.
He said: "It is a good learning experience for my team and we are happy to be at the World Cup. We made two mistakes and were punished for them. There was not enough urgency in our play."