Spanish second seed Nadal was able to see off his spirited 69th-ranked opponent 7-5 6-2 7-5, BBC Sport reported.
Playing on an empty Rod Laver Arena after Melbourne entered lockdown, Nadal was not at his rhythmic best but he held off a confident-looking Norrie.
The 20-time Grand Slam champion will play Italy's Fabio Fognini next.
"I combined very good games with a couple of unforced errors that I need to fix," Nadal said.
"That is normal and hopefully that will help me for what's coming on Monday," Nadal added.
Nadal has yet to drop a set in Melbourne despite being, by his own admission, not near his best.
Norrie played well, saving six of the 12 break points he faced, but although he would get to 0-30 on the Nadal serve he could not find a way through.
Hope for crowd return
Up to 30,000 fans a day were allowed in Melbourne Park for the first five days of the Grand Slam, before the state of Victoria went into lockdown for five days from Saturday.
Speaking about playing in front of no fans, Nadal said, "The feeling is completely different. Nobody wants this, but it's not the time to forget about the situation we're facing.
"Australia is one of the best examples in the world [for their COVID-19 response] so maximum respect for what they have been doing and hopefully the situation will improve and the crowd can come back before the end."
World number one and home favorite Ashleigh Barty, who did not play at the tournaments last year that were played behind closed doors as she did not travel because of the coronavirus pandemic, said she quite enjoyed the silence of the stadium.
"You can narrow your focus in to listen to the sound of the ball," Barty explained after her win over Ekaterina Alexandrova.
Daniil Medvedev's coach walked out as the Russian blew a two-set lead before winning a five-setter for the first time in the Australian Open third round.
The fourth seed eventually beat Filip Krajinovic 6-3, 6-3, 4-6, 3-6, 6-0.
Medvedev repeatedly yelled at his box as he went 4-1 down in the fourth – with coach Gilles Cervara eventually picking up his bag and leaving.
He did not return to see Medvedev dial back in to dominate the deciding set.
"He said just before leaving that he's sure I'm going to win the match, but he's going to leave to let me be more calm," Medvedev said of Cervara's exit.
"Sometimes maybe I will disagree but this time for sure it was a good thing to do.
"Today it helped, and definitely we're going to talk about it a little bit, but there is not a big deal."
Medvedev, among the favorites to challenge defending champion Novak Djokovic, yelled in French, Russian and English as the match against the Serb slipped from his grasp.
He also complained about the spider-camera at Melbourne Park, saying it was obstructing his toss on serve.
Medvedev, who won the season-ending ATP Finals in 2020, has often caught attention for his on-court antics.
He was booed throughout the US Open in 2019 where he reached the final, and he later admitted that the crowd spurred him on.
"Sometimes I'm a very temperamental person on the court so it can still get out, and usually it doesn't help me to play good," he said.
"Now I think I have made big steps already working on my mental strengths."
Medvedev will play Mackenzie McDonald next after the American beat Lloyd Harris of South Africa 7-6 (9-7), 6-1, 6-4.
Greek fifth seed Stefanos Tsitsipas also advanced, beating Mikael Ymer of Sweden 6-4 6-1 6-1.