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 of France, who was eliminated by Chile's Cristian Garin 2-6, 6-4, 6-5.
The tournament's top seeds, number one Casper Ruud of Norway and number two Hubert Hurkacz of Poland advanced in straight sets.
Ruud defeated Botic van de Zandschulp of the Netherlands 6-1, 6-2, and Hurkacz topped Great Britain's Jan Choinski 7-6 (5), 6-4 to reach the quarterfinals.
Other winners were Richard Gasquet of France, Spaniards Pedro Martinez and Pablo Llamas Ruiz and Hungary's Marton Fuscovics.
Grand Prix Hassan II
Roberto Carballes Baena defeated seventh seed Dan Evans 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 of Argentina fought off Switzerland's Stan Wawrinka 3-6, 7-5, 6-2 to advance to the quarter-final round.
Also moving on was American qualifier Nicolas Moreno De Alboran, who beat Belgium's David Goffin 6-3, 6-4, as well as Aleksander Vukic, who edged Sebastian Ofner to complete the quarter-final bracket.
