The South Americans reached uncharted territory after overcoming a Japan side who were also looking to make it through to the last eight for the first time in Pretoria last night.
There was nothing to separate the sides after 120 minutes in which neither nation could produce a goal or much in the way of chances, and it was left to the first penalty shoot-out of the 2010 FIFA World Cup to decide the winner.
That honour went to Paraguay, who scored all five of their spot-kicks while Japan missed their third when Yuichi Komano smashed his effort against the crossbar.
Santa Cruz was understandably delighted with the historic win, but wants their record-breaking run to continue beyond their quarter-final showdown with Spain.
"We're very pleased, very happy. It can't be measured yet how big an achievement this is for us but we are very happy to get through," Santa Cruz said.
"We set our goals very clearly and we said anything below this (reaching the quarter-finals) would be a disappointment. We've achieved that goal, but now that we are here we want to get even further and keep making history for our country.
"We are very pleased with the result, but we are also very keen to go back and rest now and have a great game in the next round."
Gerardo Martino's side will now meet Spain in Johannesburg on Saturday after the European champions knocked out Portugal.
"We are very confident that in tough games we always play better, and hopefully it will be the same again in the quarter-finals," Santa Cruz added
Paraguay were the lowest scorers of the eight pool winners after netting just three goals in their three Group F games - two of which came in the win over Slovakia.
The Albirroja also failed to create many goalscoring chances against Japan, but Santa Cruz hopes it will be a different story against the Spanish.
"We had two games where people were very concerned about the defending and it's very tough to create chances when people are defending with 10 or 11 players. We still had chances but we didn't have the fortune to score," said Santa Cruz, who has yet to get off the mark in South Africa.
"I think we're getting in positions and we're doing the right things. Some day things will start to fall for us, and hopefully it will be in the quarter-finals."
Paraguay's joy was Japan's disappointment last night as the Blue Samurai missed out on their own chance to make history.
Japan had only once before reached the last 16, on home soil in 2002, and goalkeeper Eiji Kawashima hopes they can continue to make progress for the next FIFA World Cup.
"We can take confidence and have proved that the Japanese team can play much better than imagined. That's a good experience and we have to carry on for the next World Cup," said Kawashima. "I tried to go into the penalty shoot-out with confidence but it's very disappointing I couldn't save any."