Hantuchova won 6-3 5-7 6-2 after repelling a fightback from Riske, who upset 2011 Wimbledon champion Petra Kvitova to reach the fourth round at a grand slam for the first time.
"I guess the best things happen when you don't expect them," said Hantuchova.
"(I was) just fighting like crazy, really leaving everything I've got on the court, really putting my heart into it because it means so much to me.
"I guess, sometimes you keep trying, keep trying, and then you don't know why, but it just comes all together."
Hantuchova was just 19 when she made the last eight at Flushing Meadows in 2002 but had not been back to the quarter-finals since.
The 30-year-old seemed to be cruising to victory against Riske after she raced through the opening set in just 37 minutes then led 4-2 in the second.
But the American, seven years younger, broke back to level the second set at 4-4 when the match was stopped for nearly five hours because of a thunderstorm.
When play resumed, Riske pounced, winning the set and forcing a deciding third, but could not hold off Hantuchova, who saved four points in her third round clash with Julia Glushko to continue.
"She played extremely well at the end of the second set," Hantuchova said.
"I'm just really proud the way I fought back into it in the third, raised my level when I needed to.
"The serve was working really well today. I think that was the main difference."
Riske said the rain break initially helped her because it gave her time to settle her nerves on her first appearance at Arthur Ashe Stadium but said she lost her way again in the deciding third set.
"I think I totally lost all focus and missed plenty of balls that I should make with my eyes closed. It was really a mistake on me mentally-wise for doing that," she said.
"I've never been in a moment like this before. She is a seasoned player. She played ball today.
"I did all that I could for this moment and that's all I could do."
Hantuchova is one of just two unseeded players left in the women's singles draw. Her next opponent will be either world number two Victoria Azarenka or former world number one Ana Ivanovic.
Their fourth round match was postponed until Tuesday because of the inclement weather.
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