The 19-year-old Stanford junior gave the old college try in her battle with the tournament's second seed before falling 6-2, 6-4 on Stadium Court in front of 3,546 fans.
"I'm really excited about what that loss meant today, which is kind of weird to say, but it was a great experience," said Gibbs, who is the NCAA Division I champion in singles and doubles (with teammate Mallory Burdette). "I'm looking forward to the U.S. Open."
Kvitova, who is the fifth-ranked player in the WTA, can look forward to her quarterfinal match with close friend Lucie Safarova, who won earlier in the day over Jie Zheng 6-4, 6-0, before focusing on the year's final Grand Slam.
"I want to play a lot of matches here and go to the U.S. Open with a good result, with good matches in my bag," said the 6-foot Czechoslovakian, who had a first-round bye at New Haven. "That's what I want to do."
In the first set against Gibbs, she struggled a bit in the middle, but prevailed. It was more difficult in the second set for Kvitova, with her opponent hanging in and keeping it close until they reached 4-4.
"She's doing very well," Kvitova said. "She didn't have a lot of mistakes and I think it's amazing she is playing so well. She had a great experience when she played Serena (Williams) and she played me.
"I think she's very talented."
Gibbs lost to Williams, the 2012 Wimbledon champ, last month in the second round of the Stanford Classic in straight sets. However, just with the Kvitova match, Gibbs was encouraged with her ability to play well against top players in the world and will likely turn pro next year after finishing her junior year at Stanford.
"It's always been my dream to play a main draw at any Slam and the U.S. Open is even more important to me because it's on my home turf," said Gibbs, who was playing in just her third main card event. "I'm not too scared about anyone out there, given the fact I just played two top five players this summer."
Kvitova's summer has improved after she lost in the quarterfinals in the Olympics. The following week she returned to the Tour and captured her eighth career WTA title in Montreal.
Now her attention turns to Safarova, who Kvitova not only works out with, but together they go out for meals and attend the cinema.
"We practice a lot together and we are actually very good friends," Kvitova said. "We know each other very well and it will be our first meeting. It's strange. ... I think it will be a tough match."
Their relationship adds a unique wrinkle to the quarterfinal matchup.
"Of course I'm always looking forward to the next match, but it's always tough to play good friends," Kvitova said. "I think the nerves will be there for sure."
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