Victory meant Nadal set the record for the longest Davis Cup winning streak in combined singles and doubles matches with 23 in a row.
The United States could become the first team to seal their place in the semis later on Saturday if Jack Sock and Ryan Harrison beat Belgians Sander Gille and Joran Vliegen in Nashville.
Nadal had not played for Spain since the tie against India two years ago, but was back to his vintage best as he claimed his 23rd singles victory in the competition with a 6-2, 6-2, 6-3 win over Philipp Kohlschreiber of Germany, Xinhua news agency reported. "There were times when I lost a bit of control because the match was tough, I had not competed for a long time and to maintain the level of the first set was complicated for me. I feel comfortable back on the clay", Nadal said.
Germany's Tim Puetz and Jan-Lennard Struff celebrate winning their quarter final doubles match against Spain's Feliciano Lopez and Marc Lopez.More news: Yaw Addo secures Ghana's first win at Commonwealth Games
"It's a great feeling to be back, coming back from injuries is always hard, but it's great to be in front of my crowd on a very memorable day".
Italian Fognini missed two chances to take the third set against Pouille, failing to serve out at 5-4 and then spurning a break point in the 11th game.
Noah added: "It came down to a few points in the third set". Italy holds the Davis Cup record with a 29-tie unbeaten run at home from 1949-64.
The Davis Cup is usually shown on the BBC but with Great Britain not involved the option to broadcast the quarter-finals wasn't taken up.More news: 'What have I done?' - Zuma asks as corruption trial begins
On Saturday, France's Nicolas Mahut and Pierre-Hugues Herbert will face Fognini and Simone Bolelli in the doubles.
World number one Nadal had not played since he retired from his Australian Open quarterfinal against Marin Cilic in January with a leg injury but made light work of Kohlschreiber to draw five-time winners Spain level at 1-1 in the tie. David Ferrer and Phillip Kohlschreiber will play the decisive rubber if it goes to the wire in Valencia.
Lucas Pouille fired the French ahead with a laboured 6-3 6-2 4-6 3-6 6-1 win over Andreas Seppi before home favourite Fabio Fognini redressed the balance with a 6-7 (6) 6-2 6-2 6-3 defeat of Jeremy Chardy.More news: Women amateurs will play Augusta National in 2019. How did that happen?