Defending champion Bartoli, of France, said the 18th ranked Azarenka is a rising star in the game and deserved the win.
"Every single game was tough," Bartoli said. "It was hard to win some easy points because I was fighting and running so much.
"This was a very high quality game and she just was a little better than me. She was beating me easily the last three times, so there was some improvement."
Azarenka advanced to her fourth semi-final of the year. Azarenka meets No. 1 seeded Aussie Samantha Stosur, a 7-5, 3-6, 6-3 winner over No. 7 Yanina Wickmayer.
Stosur, of Brisbane, is playing in her first tournament since breaking into the top five in the world. She leads the WTA Tour with 37 wins this year.
Azarenka improved to 4-0 against Bartoli, seeded fourth and ranked 14th.
"She has improved her game," Azarenka said. "It was a very tough game. It was important to change the rhythm with her and make her move."
Azarenka, who turns 21 on Saturday, won seven consecutive games to take control of the match in the second set after falling behind 2-0.
"She started the match well, was hitting hard and making so many winners," Azarenka said. "I was trying to stay out there and find my game. It finally went my way."
Azarenka improved to 17-5 on hard courts this year. All three of her career titles have been on hard courts.
"We've had a couple of close matches and there were some where she absolutely killed me," said Stosur, who reached her sixth semi-final of the year. "The way she plays gives me problems but my game is different than the last time we played."
Third-seeded Agnieszka Radwanska of Poland beat Belarus' Maria Kirilenko, 7-5, 6-0, in another quarter-final.