"I'm a fighter at heart," she said. "I never, never give up. Whatever the score is, I'm going to try to do my best because there is always an opportunity. There is always a way. You just have to make up your mind and try to find that way."
Ivanovic, the 2008 French Open champ, was whipping forehand winners in the first set. But then her serve deserted her, as it has before, and the 13th-seeded Serb was broken in nine of her last 11 service games.
"This was the big difference because I felt like I could break her, but it was very frustrating that I was losing my serve constantly," said Ivanovic, who had eight double-faults.
Indeed, Azarenka wasn't much better, with nine double-faults of her own.
Unable to serve out the match at 5-3 in the third set, Azarenka finally clinched the victory on her fourth match point. On Wednesday, she'll face unseeded Daniela Hantuchova, who played her fourth-round match Monday.
The meeting between Azarenka and Ivanovic was the only one of four women's matches that day to be postponed to Tuesday because of rain. Asked if anything could have been done differently to keep everybody on the same schedule, Azarenka wouldn't bite. She just smiled and said, "Mother Nature could have done something."
A few hours after Azarenka made the quarterfinals, the first woman reached the semis. Fifth-seeded Li Na beat No. 24-seeded Ekaterina Makarova 6-4, 6-7 (5), 6-2 in 2 hours, 20 minutes.
Li, the 2011 French Open champ, was up a break three times in the second set, then led 3-0 in the tiebreaker, but couldn't close the victory. From 2-2 in the third, though, she won four straight games to reach her first U.S. Open semifinal.
Li joked that she'd lie down with a bag of chips to watch the night match that will determine her next opponent. No. 1 Serena Williams takes on 18th-seeded Carla Suarez Navarro.
Another Grand Slam champion couldn't win a match that went the distance. Lleyton Hewitt, making a rousing run at age 32, was two points from victory but fell 6-3, 3-6, 6-7 (3), 6-4, 7-5 to Mikhail Youzhny in the fourth round.
Hewitt won the 2001 U.S. Open title but is currently 66th after a series of foot and hip injuries. The 21st-seeded Youzhny is a two-time semifinalist at Flushing Meadows.
The 31-year-old Russian will face the winner of Tuesday's match between No. 1 Novak Djokovic and 43rd-ranked Marcel Granollers. Defending U.S. Open champ Andy Murray also plays Tuesday night, against unseeded Denis Istomin.
Azarenka, the two-time Australian Open champ, insisted she's not concerned by two straight slow starts. She also lost the first set in the third round to 26th-seeded Alize Cornet before rallying for a 6-7 (2), 6-3, 6-2 win.
She's gotten into it with the chair umpire in two straight matches as well. On Tuesday, it was over Ivanovic being awarded a winner, instead of the two replaying the point, after a delayed out call was overturned in the first set.
Azarenka looked sluggish after a late-morning start and didn't make her first winner until the sixth game. But Ivanovic has often wilted under pressure since her lone Grand Slam title more than five years ago; she hasn't been back to a major semifinal since.
Azarenka started winning points on drops shots she said she's always had that in her repertoire but just hadn't need them much before.
There were five straight breaks to open the second set. After Ivanovic double-faulted on break point for the second straight service game, Azarenka finally held to lead 4-2.
The start of the third set was much of the same. Azarenka held in the first game, then there were four straight breaks.
Azarenka preferred to joke about the two players' serving struggles: "I think we're just great returners."
Azarenka had two match points while serving at 5-3 but wasted both with backhand errors.
The way Ivanovic was serving, though, Azarenka was still in prime position to close the match, and sure enough, she broke Ivanovic again in the next game to advance.
It took 2 hours, 10 minutes, but the 2012 U.S. Open runner-up wasn't going to fret about her uneven play.
"If I have to fight and dig deep to find a way not playing at your best against a player who was playing that good," Azarenka said, "I think it's pretty remarkable to walk off the court with a win."