Verdasco came into the match on the blue clay of the Manolo Santana show court on a high after dumping out second seed Rafa Nadal on Thursday, his first victory over his Spanish compatriot in 14 attempts.
However, sixth-seeded Czech Berdych, who thumped French 12th seed Gael Monfils 6-1, 6-1 to reach the last eight, comfortably ended a two-match losing streak to Verdasco on clay to set up a semifinal against Juan Martin Del Potro.
The 2009 US Open champion, undefeated in 10 clay matches this season and winner in Estoril last week, eased to a 6-3, 6-4 victory against Ukrainian 16th seed Alexandr Dolgopolov in the earlier match on court three.
"I am really happy with my tennis, with my body and I am feeling really confident on clay," Del Potro, who had an eight-month layoff in 2010 due to a wrist injury, told a news conference.
"I don't feel any pain so it's a good sign for the next tournament and I think I'm getting better tournament by tournament," the 23-year-old added.
World No.1 Novak Djokovic, who has joined Nadal in threatening to boycott the event next year because of what they say are dangerously slippery courts, meets Serb compatriot Janko Tipsarevic, the seventh seed, later on Friday.
Victory for the defending champion will set up a semifinal clash against Swiss third seed Roger Federer, who will rise above Nadal to No. 2 two if he wins the title, or Spanish fifth seed David Ferrer.
In the women’s section, top seed Victoria Azarenka saw off China's Li Na with a battling 3-6, 6-3, 6-3 win to reach the semifinals yesterday.
The Belarusian world number one will now play either Agnieszka Radwanska, the Polish fourth seed, or American qualifier Varvara Lepchenko.
Azarenka, January's Australian Open champion, leveled her head-to-read record against French Open champion Li at 4-4 with the victory.
Eighth seed Li was playing in her second clay quarterfinal after Stuttgart.
Ninth seed Serena Williams won a battle of former number one players as she defeated second seed Maria Sharapova, the Stuttgart champion, 6-1, 6-3 with 11 aces, 28 winners and five breaks of serve in 80 minutes.
Williams said that while the blue clay courts remain problematic, the issue was not enough to make her consider skipping the event in 2012 as Rafael Nadal and Novak Djokovic have threatened to do.
"This is a tough surface and it's extremely slippery," said the 13-time Grand Slam-winner.
"But Djokovic is still in the event and so is Roger (Federer), last time I checked. If I'm not here next year, it won't be because of the clay.
"I don't think there's been any improvement in the courts over the week, but every clay court is different. This is not the best court — definitely not what they play like at Roland Garros.
