Jankovic, bidding for her first Grand Slam singles title, beat Flipkens 6-4, 7-5 on Wednesday.
The 23-year-old Serbian player lost last year's U.S. Open title to Serena Williams and also made the semifinals in last year's Australian Open and French Open and the fourth round at Wimbledon.
Ivanovic lost the last Australian Open final to Maria Sharapova before winning her first Grand Slam title at the French Open and gaining the No. 1 ranking.
"I was in this situation before French Open last year, coming into the tournament as pretty serious finalist," Ivanovic said. "That experience will help me a lot to approach this Australian Open."
Now seeded fifth, the 21-year-old Serb is getting inspiration from memories of her run here last year.
"It was amazing, I'm so happy to be back here," Ivanovic said. "I had lots of great matches here last year."
Dictating play and ripping winners into the corners, Ivanovic repeatedly took advantage of Brianti's weak serve. The 28-year-old Italian held serve only three times.
No. 169-ranked Brianti needed treatment on her lower back after getting broken to fall behind 3-0 in the second set. She returned to break Ivanovic's serve and held to pull back to 3-2 but lost the next three games.
Ivanovic, pushing the limits with her shots, had the same number of winners as unforced errors (26).
She won 17 of her 27 net approaches, a sign she's growing in confidence moving forward.
"I improved a lot since last match. I was committing much more, coming to the net more often," she said, adding she still had plenty to improve.
"When you come against top players or high-ranked players, you automatically raise your level and your consistency. That's something that I can do," she said.
Safina lost to a qualifier in the first round here last year but went on to be runner-up at the French Open, take the Olympic silver medal and reach the U.S. Open semifinals.
"I'm just trying to find my game. I'm trying to be aggressive," Safina said, explaining the erratic nature of some of her shots, before admitting that she was enjoying her status as a leading seed.
"Now when I practice there is more people standing watching me ... I get a better court, I'm not anymore just Marat's sister," she grinned.
Unseeded Jelena Dokic, a former Wimbledon semifinalist making her first appearance in the main draw of a Grand Slam after a three-year absence to deal with personal problems, reached the third round by ousting 17th-seeded Anna Chakvetadze 6-4, 6-7 (4), 6-3.
Information from The Associated Press and Reuters was used in this report.