Sixth seed Nadal, the only former Melbourne champion left in the men's draw, beat American Giron 6-1 6-4 6-2, BBC Sport reported.
The Spaniard is making his return to Grand Slam tennis after a five-month layoff with a foot injury.
"I have had very challenging months, tough moments and a lot of doubts," he said on court.
Nadal is the only member of the 'Big Three' competing in Melbourne after nine-time Melbourne champion Novak Djokovic's visa was cancelled on Sunday.
His sole Australian Open title came in 2009 but he won the Melbourne Summer Set in the build-up to the tournament.
He was rarely troubled against Giron, hitting 34 winners to help set up a second-round meeting with Germany's Yannick Hanfmann, who beat home favourite Thanasi Kokkinakis in straight sets.
"You never know when you come back from injuries – things are difficult and you go day by day," Nadal, 35, said.
"But I'm staying positive. I won three matches and a title here last week."
Elsewhere, Wimbledon finalist Matteo Berrettini overcame stomach problems to beat American 20-year-old Brandon Nakashima.
The Italian took a medical timeout during the second set and wrote 'Imodium, grazie' on the camera lens after his 4-6 6-2 7-6 (7-5) 6-3 victory.
"I've been feeling sick with my stomach, let's say that," Berrettini said.
"I really held with my mental strength mostly, because my body wasn't there."
Canadian 14th seed Denis Shapovalov beat Serbia's Laslo Djere 7-6 (7-3) 6-4 3-6 7-6 (7-3) and Polish 10th seed Hubert Hurkacz advanced with a 6-2 7-6 (7-3) 6-7 (5-7) 6-3 victory over Egor Gerasimov of Belarus.
Women’s defending champion Naomi Osaka overcame an unexpectedly stiff challenge from Colombian Camila Osorio to reach the second round of the Australian Open 6-3 6-3 on Rod Laver Arena on Monday.
The Japanese former world number one looked all business as she raced to a 5-0 lead in the opening set but a first hold of serve invigorated Osorio, who then started chasing down everything Osaka threw at her, Reuters reported.
Osaka needed to save two break points to hold serve for the first set and faced two more in the second stanza before overpowering the inexperienced world number 50 with her aggressive shots from the baseline.
"It always feels special to come back here," said the 13th seed, who will next face American Madison Brengle as she seeks her third Melbourne Park title.
"I thought she played amazing. Overall I'm just happy to be here, I'm happy to see everybody in the audience and I hope we gave you a great performance."
Australian Prime Minister Scott Morrison has left the door open for Djokovic to compete at next year's Australian Open despite the tennis superstar facing an automatic three-year ban from entering the country.
The world number one player left Australia late on Sunday after the Federal Court upheld a government decision to cancel his visa, capping days of drama over the country's COVID-19 entry rules and his unvaccinated status, Reuters reported.
Under immigration law, Djokovic cannot be granted another visa for three years unless Australia's immigration minister accepts there are compelling or compassionate reasons.
"I'm not going to precondition any of that or say anything that would not enable the minister to make the various calls he has to make," Morrison told 2GB radio on Monday as Djokovic was en route to Dubai.
"It does go over a three-year period, but there is the opportunity for (a person) to return in the right circumstances, and that will be considered at the time."
The unanimous ruling by a three-judge Federal Court bench dealt a final blow to Djokovic's hopes of chasing a record 21st Grand Slam win.
Immigration Minister Alex Hawke had said Djokovic could be a threat to public order because his presence would encourage anti-vaccination sentiment amidst Australia's worst coronavirus outbreak.