Fifth-seeded Ivanovic was broken once in each set but fended off five other break points before finishing with consecutive aces against Goerges, a German ranked No. 107 and playing only her fourth major.
Ivanovic lost the 2008 Australian final to Maria Sharapova, who is not defending her title because of an injury, and then won the French Open to move into the No. 1 ranking midway through the year.
But the 21-year-old Serb was bothered by injuries in the middle of the season and her form at the majors waned, going out in the third round at Wimbledon and the second round at the U.S. Open.
She and Goerges finished with a combined 66 unforced errors to just 34 winners and were called for several foot faults. Ivanovic was pleased to have escaped with a victory.
"I was so happy to be back," she said.
Seventh-seeded Andy Roddick was more thorough in a 6-0, 6-2, 6-2 win over 31-year-old Swedish qualifier Bjorn Rehnquist in the first match at Rod Laver Arena.
The 2003 U.S. Open champion lost only nine points in the first set, sealing it in 28 minutes after setting up triple set point with consecutive aces.
Roddick, a semifinalist in Australia every alternate year since 2003, had 17 winners, nine aces and kept his unforced errors down to 10.
He came into the tournament as runner-up to fourth-ranked Andy Murray at the Qatar Open in Doha and hasn't given up hope of winning a second major, despite the domination of Roger Federer and Rafael Nadal.
"I'm 26, so I'm not quite over the hill," he said. Another Grand Slam title "is what I'm shooting for."
Federer, bidding to equal Pete Sampras' record 14 major titles, had a night match later Monday against Andreas Seppi of Italy.
Third-ranked Novak Djokovic, the defending champion, will play Andrea Stoppini in the third match on center court, following fellow Serb and women's No. 1 Jelena Jankovic's match against Yvonne Meusburger.
In other men's results, 20th-ranked Tomas Berdych of the Czech Republic beat American Robby Ginepri 6-4, 6-4, 6-3.
Women's No. 7 Vera Zvonareva of Russia posted a 7-6 (2), 6-0 win over Slovakia's Magdalena Rybarikova and No. 15 Alize Cornet of France beat Ukraine's Mariya Koryttseva 6-3, 6-4.
Former fourth-ranked Kimiko Date Krumm returned to the main draw of a major after a 13-year hiatus. The 38-year-old Japanese player came through qualifying on the weekend and faced Estonia's Kaia Kanepi, seeded 25th, on Court 6.
On a warm, cloudless day, Ivanovic constantly struggled to find her form against Goerges, committing 10 unforced errors and three double-faults in the first five games alone.
With the midday sun playing havoc on serve tosses at one end, the errors continued to pile up as the two players swapped breaks in the first two games of the second set Ivanovic double-faulted on break point.
After another double-fault, she shouted and pumped her fist after she fended off double break point while serving at 2-2, then saved another pair of Goerges break chances at 3-3.
Ivanovic then made Goerges pay for her lost opportunities, breaking in the next game for a 5-3 advantage. She slammed back-to-back aces her second and third to advance to the second round.