Rinderknech, who is ranked exactly 100th in the world, picked up the biggest win of his career in style as he dispatched of second seed Agut with a comfortable 6-2 6-4 victory
In another upset, world number 155 Hugo Gaston laid down an early marker with a first-set demolition of Delbonis, who lost out to eventual winner Pablo Carreno Busta in the Hamburg European Open semi-finals last week, before he triumphed in three sets, 6-0 6-7 (2-7) 7-6 (7-3).
Fourth seed Cristian Garin cruised past Marc Polmans 6-3 6-0 to wrap the match up in less than an hour and seal his fifth ATP quarter-final of 2021.
Another semi-finalist in Hamburg, Laslo Djere survived a second-set scare to eliminate Zizou Bergs with a 6-2 4-6 6-2 win, which sets up a last-eight tie with Rinderknech.
At the Croatia Open in Umag, fourth seed Richard Gasquet battled past qualifier Alessandro Giannessi to secure a 7-6 (7-4) 6-4 victory.
Top seed Albert Ramos-Vinolas also progressed, winning 7-5 6-4 to eliminate Duje Ajdukovic on his home turf.