Tiafoe, the second seed, got past Russia's Aslan Karatsev 7-6 (5), 7-6 (5) in second-round action. Neither player broke serve in the match, with Tiafoe saving four break points and Karatsev saving two.
In the first-set tiebreaker, Tiafoe jumped out to a 4-1 lead before prevailing on his third set point. In the second-set tiebreaker, Karatsev led 5-3 before Tiafoe won the final four points.
Nakashima edged Australia's Aleksandar Vukic 7-6 (5), 7-6 (9) in a first-round match. Vukic had two set points in the second-set tiebreaker before Nakashima wrapped it up on his third match point.
Russia's Alexander Shevchenko (22) knocked out eighth-seeded Sebastian Korda of the U.S. 5-7, 7-6 (4), 6-4 in a second-round match.
Other second-round winners were fifth-seeded Grigor Dimitrov of Bulgaria, Chinese qualifier Juncheng Shang, Finland's Emil Ruusuvuori and the United States' Michael Mmoh.
Japanese lucky loser Sho Shimabukuro, U.S. qualifier Zachary Svajda and Australians Thanasi Kokkinakis and Jordan Thompson earned first-round victories.
Generali Open
Eighth-seeded Daniel Altmaier of Germany rallied for a 6-7 (5), 6-2, 6-4 win over Switzerland's Marc-Andrea Huesler in the first round at Kitzbuhel, Austria.
Altmaier came through in the match's key moments, saving three of the four break points that he faced and converting all four of his break opportunities.
Fifth-seeded Laslo Djere of Serbia downed Australia's Christopher O'Connell 6-4, 7-6 (4), but seventh-seeded Serbian Dusan Lajovic lost 6-3, 6-2 to China's Zhizhen Zhang. Austria's Dominic Thiem, Brazil's Thiago Seyboth Wild and France's Arthur Rinderknech also logged opening-round wins.
Mifel Tennis Open
Fifth-seeded Alex de Minaur of Australia overcame an inconsistent serving performance to beat Tunisia's Skander Mansouri 6-4, 6-4 in the first round at Los Cabos, Mexico.
de Minaur put just 47 percent of his first serves in play, but those serves were effective when they were accurate, as he won 22 of 26 points. He failed to save any of the three break points on his serve, but he broke Mansouri's serve on five of 11 opportunities.
Seventh-seeded Dominik Koepfer of Germany cruised to a 6-4, 6-3 victory over Chile's Tomas Barrios Vera. Also winning their openers were sixth-seeded Nicolas Jarry of Chile, Belarus' Ilya Ivashka, Croatia's Borna Gojo, Australia's James Duckworth and Taiwanese lucky loser Jason Jung.
