Having spent the latter part of 2014 recovering from illness and injury, the Spanish third seed now finds himself in the last eight, needing two hours to beat Anderson at Rod Laver Arena.
Nadal next faces Czech seventh seed Tomas Berdych, who he has beaten 18 straight times, for a semi-final berth.
"(Berdych) is playing great. Will be a very tough opponent, no?" Nadal told reporters.
"But for me, quarter-finals is a great result, talking seriously. Arriving here, losing in the first round of Qatar, not playing matches for the last seven months, to have the chance to be in quarter-finals again here is a very positive thing for me.
"I try to play better and better every day. If that happens, I hope to keep having chances for the next match.
"But today is a day to be happy the way that I improved my level of everything… all the things I have to do on court."
Nadal faced some tough serving from the 14th seeded Anderson in the first set to eventually prevail 7-5.
After the first set Nadal went up a gear and took control of the match.
"So (I) just tried to play with the right decision, right determination, and I did. Worked well. But sometimes works well, sometimes doesn't," Nadal said.