The 24-year-old second seed from Belarus took advantage of Ivanovic's 42 unforced errors and broke the former world number one nine times in the match, winning despite surrendering seven breaks herself and hitting just 21 winners.
Ivanovic had fired 15 forehand winners before Azarenka managed her first.
"Ana just came out firing. She played really amazing tennis in the first set," Azarenka said. "I was sleeping and she was ripping winners.
Azarenka, who lost to Serena Williams in last year's US Open final, will play for a semi-final spot against Slovakia's 48th-ranked Daniela Hantuchova, with whom she has split four career meetings.
"She's playing the best tennis of her life," Azarenka said of Hantuchova. "She is really playing well and has a lot of confidence. We hit with each other a lot so there will be no surprises there."
Azarenka, a semi-finalist in five of the past seven Grand Slams, defeated world number one Williams in the final at Cincinnati in the last big US Open tune-up event to become the main threat to dethrone Williams.
Neither has dropped a set in reaching the last eight.
Ivanovic, the 2008 French Open winner, made her first Slam quarter-final since her Roland Garros crown at last year's US Open, but could not duplicate the effort.
After Ivanovic took an early break and served for the first set, Azarenka broke back in the ninth game, using an overhead smash on her third chance, to begin a run of seven consecutive broken serves.
Azarenka broke for a 2-0 lead in the third set but Ivanovic answered with a lob winner to break back in the third game and they exchanged breaks twice more before Ivanovic held to pull level at 3-3.
Azarenka held and Ivanovic was broken again, her backhand wide allowed the Belarussian to serve for the match.
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