Yet she dropped just one service game against No. 34-ranked Giorgi, who struggled with 12 double-faults, including one on the decisive break point in the 11th game of the second set.
Williams fired two aces and a service winner in the last game.
"I haven't played a competitive match in a really long time, but I think it was really good for me to come out here," Williams said. "I was able to stay in it and stay calm today and I think that's what matters most."
Williams said she needed the break to recharge at the end of last season and felt OK about her limited preparation.
"I haven't played in a long time, but I have been playing for 30 years, so it's kind of -- I try to focus on that," she said.
Two-time Wimbledon champion Petra Kvitova earned the distinction of winning the first completed match at the tournament when she beat Thai qualifier Luksika Kumkhum, avenging her upset three-set loss in the first round here in 2014.
Kvitova, who had to withdraw from a warmup tournament in China because of a stomach virus, said her preparation was disrupted and she was nervous ahead of the rematch with Kumkhum.
"Now I can breathe a little bit normally," she said. "I got really nervous before the match. I didn't sleep well actually. The memories from two years ago were a little bit difficult."
No. 24 Sloane Stephens, who beat Serena Williams in the 2013 Australian Open quarterfinals, lost 6-3, 6-3 to Chinese qualifier Wang Qiang, while No. 26 Anastasia Pavlyuchenkova and No. 27 Anna Karolina Schmiedlova also went out in the first round.
Djokovic's bid for a sixth Australian Open title started in temperatures of 34 Celsius (93 Fahrenheit), and he became annoyed with himself and his play at times, including three double-faults.
Overall, though, he was consistently too good for one of the up-and-coming players on tour. He said everything started looking up for him when he became a dad back in 2014, then joked about maybe having another baby.
Sixth-seeded Tomas Berdych, a semifinalist last year, opened his 50th consecutive major by beating Yuki Bhambri of India 7-5, 6-1, 6-2, and No. 7 Kei Nishikori had a 6-4, 6-3, 6-3 win over Philipp Kohlschreiber.
Also advancing were No. 12 Marin Cilic, No. 15 David Goffin, No. 19. Dominic Thiem and No. 26 Guillermo Garcia-Lopez. No. 22 Ivo Karlovic was the first of the men's seeds to exit the tournament when he retired due to a left knee injury while trailing Federico Delbonis 7-6 (4), 6-4, 2-1.
-AAP