Roddick hopes to face Spain in his hometown of Austin, Texas
Andy Roddick secured the United States a Davis Cup quarter-final with Spain in July by giving them a 3-1 lead against Chile after defeating Paul Capdeville.
Roddick, a former world number one, came from behind to win 3-6 7-6 (7-2) 6-3 6-3 on the clay in Santiago.
Holders Serbia also went through with a 4-1 victory over India, Viktor Troicki completing the win by defeating Somdev Devvarman 6-4 6-2 7-5.
Serbia will now face Sweden, who beat Russia 3-2.
Serbia were without Novak Djokovic, who skipped the tie to concentrate on the Indian Wells event which begins this week, but Troicki is hopeful they can call on the world number three when they travel to Sweden.
"It is a great feeling to be in the last eight of the competition because this means we have stayed in the World Group, while it also gives us a chance to carry on with our title defence," Troicki said.
WORLD GROUP 1st ROUND
Serbia 4-1 India
Sweden 3-2 Russia
Czech Rep 2-3 Kazakhstan
Argentina 4-1 Romania
Chile 1-4 USA
Belgium 1-4 Spain
Croatia 2-3 Germany
Austria 2-3 France
"We are through to the quarter-finals, we know it's going to be very tough against Sweden and we hope to have our best player Novak Djokovic back for that tie."
Roddick's win means the USA's quarter-final with Spain, who beat Belgium this weekend, could take place in Roddick's hometown of Austin, Texas, with San Antonio and Albany also being considered.
"I think it's been no secret that I've wanted to play in Austin, so I certainly appreciate the effort put forward and hopefully we can get it," Roddick said. "It would be a dream come true to play at home, a really cool experience.
"A guy named Rafa coming to town will provide a little bit of energy."
Germany also progressed courtesy of late replacement Philipp Petzschner's enthralling 6-4 7-6 7-6 victory over Croatia's Ivo Karlovic to give Germany a 3-2 win.
Germany will face France in the quarter-finals after Jeremy Chardy recovered from a set down to defeat Martin Fischer of Austria 2-6 7-6 6-3 6-3 and wrap up a 3-2 victory.
"The important thing is to advance. Next time maybe it will be another who will get two points," said Chardy, who also achieved France's first point by beating world number 10 Jurgen Melzer in Friday's opening match.
Meanwhile, Romania were thrashed 4-1 by Argentina in Buenos Aires after wins for David Nalbandian, Juan Monaco, Eduardo Schwank, and Schwank and Juan Ignacio Chela in the doubles.
Argentina will face Kazakhstan in the quarter-finals following their 3-2 victory over the Czech Republic, a tie that was decided in the fifth and deciding rubber, with Mikhail Kukushkin defeating Jan Hajek 4-6 6-7 7-6 6-0.