Clijsters, who won the Cincinnati final last week in three sets over Maria Sharapova, had to outlast American Bethanie Mattek-Sands the night before.
"I was playing an opponent who could not miss a shot, while today Kanepi was missing a lot, especially since her strengths are usually her serve and her backhand," Clijsters said. "I really felt that I had opportunities once I was in a rally with her, and she was either going to make a mistake or I could really try to make her move.
"It's always nice to win a tournament and gradually feel your level improve and also to have a matches like (Mattek-Sands) and in Cincinnati _ to be behind and fight back and win."
Clijsters will face Vera Zvonareva, the eighth seed from Russia who downed Agnes Szavay of Hungary 6-3, 6-3.
Svetlana Kuznetsova had a 6-4, 1-6, 6-3 victory over No.7 Agnieszka Radwanska that put her into the quarterfinals, turning around the match after getting thrashed in the second set.
So what did she do during the break?
"I take a vodka shot, no?" Kuznetsova said with a laugh. "No, nothing like that. ... I just go to the locker-room, change all my clothes, eat a banana and I go back. It's very simple. It was 6-1, but almost every game was deuce, so it was pretty hard. I took a break and it worked."
The win put 11th-seeded Kuznetsova into a quarterfinal Friday against Zheng Jie of China, who upset fourth-seeded Elena Dementieva 7-6 (3), 6-4.
The weather turned cool for the evening session after a brief rain delay, but it didn't stop second-seeded Carolina Wozniacki from defeating 15th-seeded Flavia Pennetta 4-6, 6-3, 6-1 for her fourth win in as many meetings with the Italian.
Wozniacki next plays the winner of a late Thursday match between French Open champion Francesca Schiavone, seeded sixth, and 2008 Rogers Cup champ Dinara Safina.
In other matches, 10th-seeded Victoria Azarenka downed ninth-seeded Li Na of China 6-3, 6-3, and 17th-seeded Marion Bartoli of France ousted Czech qualifier Iveta Benesova 6-0, 6-1, who was coming off an upset win over top-seeded Jelena Jankovic.
Azarenka and Bartoli will face off in the quarterfinals.
It has been a difficult year for Kuznetsova, who was the first Russian woman to win the U.S. Open in 2004 and added a French Open title last year.
Back at Roland Garros as defending champion in May, Kuznetsova was beaten in the third round and the massive loss of points dropped her to 19th in the world rankings. Then she went down in the second round at Wimbledon, among other dubious results in 2010.
When she crossed to North America for the hardcourt season, things seemed to turn around. It was capped by a tournament win in San Diego two weeks ago, when she beat Radwanska in the final.
"I'm not used to having (played) so few matches," Kuznetsova said. "It's hard because the other players had so many matches from the start of the season."
Her promising form may also bode well for the U.S. Open, which begins Aug. 31 in New York. As well as her victory, she also was reached the final in 2007, losing to Justine Henin, who isn't at the Rogers Cup. In her career, Kuznetsova is 2-16 against Henin.
"I love to play in the U.S. Open," she said. "It's good when you get there and you know that at least you have a game and you know what you're doing on the court."
Clijsters only loss to Zvonareva was the last time they played _ in the Wimbledon quarterfinals this year.
"We always had tough matches, a lot of three-setters or very tough ones," Zvonareva said. "I never managed to beat her and finally, at Wimbledon, I was able to play consistently.
"I always believe I could beat anyone on the other side of the net. You have to come out there and not only know the game plan, but be able to execute it. It is very important to play your best to beat someone in the quarter-finals in such a big event."