Despite playing a marathon five-setter the day before, 121st ranked Daniel looked the fresher of the pair at the outset of the match, breaking Nadal to take a 5-4 advantage when the Spaniard dumped a forehand into the net.
He eventually won the set and appeared poised to deliver a massive upset.
But Nadal soon found the accuracy that had eluded him in the first set and his booming serve proved too much for the exhausted 24-year-old, who is now 0-6 against top 20 opponents in his career.
"All the matches are difficult, especially here where everyone wants to play their best," Nadal said in an on-court interview at the Arthur Ashe Stadium after the win.
"It's true that I didn't play very well tonight but it's also true that I'm through to the next round."
Nadal is now one step closer to a potential semi-finals match against third seed Roger Federer, who won his second consecutive five-set match earlier on Thursday.
"I am here to improve every day. I believe I can do it much better and I really feel I am going to do it much better," Nadal said.
Next up for the 31-year-old is a third round match against aggressive Argentinian Leonardo Mayer.
(Reporting by Rory Carroll; Editing by Amlan Chakraborty)
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