Gaston converted three of four break points and finished with a 36-26 edge in winners, including a 20-8 advantage in the third set.
Joining de Minaur on the sidelines were second seed Stefanos Tsitsipas of Greece and third seed Felix Auger-Aliassime of Canada. Czech fifth seed Jiri Lehecka swept past Tsitsipas 6-4, 6-4, while Spaniard Roberto Bautista Agut rallied for a 3-6, 6-2, 7-6(6) victory over Auger-Aliassime, despite the Canadian firing 19 aces and saving three match points.
Gaston will face Bautista Agut in the semi-finals, while Lehecka will take on American eighth seed Marcos Giron, a 6-2, 6-2 winner over Belgium's Zizou Bergs.
BNP Paribas Nordic Open
Three-time major winner Stan Wawrinka of Switzerland drew another step closer to an elusive tournament title when he felled No. 1 seed Andrey Rublev of Russia 7-6(5), 7-6(5) in the quarter-finals in Stockholm.
Wawrinka, 39, hasn't won on the ATP Tour since May 2017. He is the third-oldest semi-finalist at an ATP event since 1990. His age didn't stop him from firing 14 aces past Rublev and saving three of four break points.
Wawrinka's semi-final opponent will be American No. 4 seed Tommy Paul, who beat Serbia's Miomir Kecmanovic 7-6(9), 6-2. Dutch seventh seed Tallon Griekspoor upset Norwegian No. 2 seed Casper Ruud 7-5, 7-6(5), and third seed Grigor Dimitrov of Bulgaria beat Switzerland's Dominic Stricker 6-3, 6-2.
Almaty Open
Aleksandar Vukic of Australia fended off No. 1 seed Frances Tiafoe 6-2, 7-6(11) in the quarter-finals in Kazakhstan.
Vukic, who had 13 aces among his 35 winners, trailed the second-set tiebreaker 4-1 before charging back. Tiafoe then saved four match points and held an 11-10 lead, but Vukic struck the final blow with the last three points of the match.
Vukic is set to face Russian third seed Karen Khachanov, who beat Kazakh wildcard Beibit Zhukayev 6-2, 6-4. No. 4 seed Francisco Cerundolo of Argentina breezed past Kazakhstan's Alexander Shevchenko 6-2, 6-3 in 64 minutes, but Canada's Gabriel Diallo knocked out second seed Alejandro Tabilo of Chile 3-6, 6-3, 6-4.