Nadal was made to work harder in the second where the players exchanged breaks over six consecutive games before the world number one finally held in the ninth to seal the match when Paire sent a service return wide. "That's what I needed, a match like this to be a little more confident".
"So, I can call it the best moment of my life".
Tsitsipas set up match point with a scorching cross-court victor before completing the victory in 2 hours, 17 minutes. "I knew I was playing pretty good today".
Nadal will face Stan Wawrinka in the round of 16 after the Swiss rallied to beat Hungarian Marton Fucsovics 1-6 7-6 (2) 7-6 (10).
Garcia proved too strong for Sharapova with a 6-3, 6-2 win and will take on top seed Simona Halep of Romania in the quarter-finals of the Women's Tennis Association Premier 5 tournament.
Heavy morning rain delayed the match by about an hour.
"It's true that we had a clinker", he told Tennis World, "but time has gone since when we lift the spirits".
Also, Karen Khachanov of Russian Federation beat 12th-seeded Pablo Carreno Busta of Spain 6-4, 7-6 (3), and Ilya Ivashka of Belarus topped American Ryan Harrison 7-6 (5), 6-4.
"But now I'm more familiar with his game. I didn't play well".
Tsitsipas struggled with a first-serve percentage of 47 percent, well below Zverev at 70 percent. But the teenager converted all four of his break-point opportunities while Zverev was five for 14.
Pointing out that Cincinnati remains the only Masters 1000 event which he did not win in his career, the 31-year-old said that he is looking forward to go there and put his best foot forward. (The early break in the third set) was everything.
Elsewhere, veteran Dutchman Robin Haase advanced to the quarter-finals for the second consecutive year, beating Denis Shapovalov 7-5 6-2. He rose to prominence previous year with a stellar run at the Rogers Cup in Montreal that ended with a semifinal loss to eventual champion Alex Zverev.
Toronto's Daniel Nestor - the oldest player at the tournament at 45 - and Vancouver's Vasek Pospisil were scheduled to play their first-round doubles match in the late afternoon.