Haas also became the oldest man to reach the quarters at any major event since Andre Agassi at the 2005 U.S. Open.
Seeded No. 12, Haas needed 13 match points to beat John Isner in a third-round marathon, but the victory over Youzhny took less than 90 minutes. By the second set Youzhny was so frustrated he demolished his racket by banging it nine times against his changeover chair.
"It's bad luck for that racket," Youzhny said.
Haas served well and dominated with his returns, winning 25 of 41 points on Youzhny's first serve and breaking eight times.
"I give him too many chances play well," Youzhny said. "That's why he play well."
Victoria Azarenka swept the final nine games and advanced to the women's quarterfinals by beating 2010 champion Francesca Schiavone 6-3, 6-0.
Seeded No. 3, Azarenka improved to 26-2 in 2013. She has won the Australian Open each of the past two years, and Roland Garros is the only major tournament where she has yet to reach the semifinals.
Haas became a first-time Roland Garros quarterfinalist in his 12th appearance at the tournament, a record for such a breakthrough. He's the first German man to reach the final eight since Michael Stich and Bernd Karbacher in 1996.
He's the oldest men's quarterfinalist at Roland Garros since 39-year-old Istvan Gulyas in 1971.
Ranked No. 2 when he was 24, Haas is now the oldest player in the top 100. He was sidelined from February 2010 to April 2011 because of hip and shoulder injuries, but he's 25-9 this year, and his ranking his risen to 14th from 205th at the beginning of 2012.
His next opponent will be No. 1 Novak Djokovic or No. 16 Philipp Kohlschreiber.
Play began in weather so cool both players dressed in long sleeves, and Azarenka wore leggings under her skirt. From the baseline the two women traded strokes and grunts.
"Ah-heee!" Azarenka shrieked.
"Wah-hooo!" Schiavone responded.
There were four consecutive service breaks before Azarenka took the lead for good, holding to make it 4-3. The superior pace and depth of her shots began to wear on the unseeded Schiavone, and Azarenka took charge by winning 11 of 12 points during a stretch late in the first set.
The Belarusian broke serve seven times and committed only 14 unforced errors.
"It was definitely a very good challenge for me to play against Francesca, especially knowing how well she's done here in the past," Azarenka said. "I'm glad with the way I played today."
Azarenka is 0-2 in French Open quarterfinals. Her opponent this time will be the winner of the fourth-round match between No. 12 Maria Kirilenko and unseeded Bethanie Mattek-Sands.
"The next step is quarterfinals," Azarenka said. "Never been past it here, so it would definitely be a much better result."