Borges, who is seeking his first ATP Tour win, saved two set points serving at 5-6 in the first set during the one-hour, 37-minute match on clay courts. Down 3-1 to open the match, Borges rallied to set up the tiebreaker.
Also upset was fifth seed Arthur Fils (19) of France, who was eliminated by Chile's Cristian Garin (27) 2-6, 6-4, 6-5.
The tournament's top seeds, number one Casper Ruud (25) of Norway and number two Hubert Hurkacz (27) of Poland advanced in straight sets.
Ruud defeated Botic van de Zandschulp (28) of the Netherlands 6-1, 6-2, and Hurkacz topped Great Britain's Jan Choinski (27) 7-6 (5), 6-4 to reach the quarterfinals.
Other winners were Richard Gasquet (37) of France, Spaniards Pedro Martinez (26) and Pablo Llamas Ruiz (21) and Hungary's Marton Fuscovics (32).
Grand Prix Hassan II
Roberto Carballes Baena (31) defeated seventh seed Dan Evans (33) of Great Britain 6-4, 7-6 (3) in the second round in Marrakech, Morocco.
The Spaniard scored 72 per cent of his points on his first serve and also fought off six of seven break points.
Mariano Navone (23) of Argentina fought off Switzerland's Stan Wawrinka (39) 3-6, 7-5, 6-2 to advance to the quarter-final round.
Also moving on was American qualifier Nicolas Moreno De Alboran (26), who beat Belgium's David Goffin (33) 6-3, 6-4, as well as Aleksander Vukic (27), who edged Sebastian Ofner (27) to complete the quarter-final bracket.
U.S. Men's Clay Court Championship
U.S. players emerged with a 2-1 edge against Australian opponents in the second round at Houston.
Third-seeded Frances Tiafoe outlasted James Duckworth 6-2, 6-7(4), 7-5, and Michael Mmoh upset eighth-seeded Max Purcell 6-2, 7-5.
In the last match of the night, sixth-seeded Aussie Jordan Thompson edged Aleksandar Kovacevic 4-6, 7-6(1), 7-6(7). Kovacevic had a match point at 5-6 in the second set, then two more in the third-set tiebreaker before Thompson converted his first match point.
Fourth-seeded Tomas Martin Etcheverry also advanced to the quarterfinals with a 6-4, 6-4 victory over Colombia's Daniel Elahi Galan
