The 12th-seeded Spaniard posted a 7-6 (2), 7-6 (4), 7-6 (3) win over Shapovalov, the 18-year-old Canadian, in a two-hour, 54-minute match.
Carreno Busta will face either Frenchman Lucas Pouille or Argentine Diego Schwartzman in the quarterfinals.
"It's amazing. It was the first time I played singles on this court," Carreno Busta said in a post-match interview. "Arthur Ashe (Stadium) is unbelievable. I cannot describe it. I tried to fight all the time, every point. You never know when your opponent will doubt a little bit or make a mistake. I was focused on my game and I got through."
This will mark Carreno Busta's second major quarterfinal appearance of the year. He also reached the quarters in Paris at the French Open.
Shapovalov had 54 winners in defeat.