Delbonis lost only seven points on serve in a 6-4, 6-3 win over Dmitry Tursunov, who is 46 spots higher in the world rankings. Four of the serving points Delbonis lost came in a second-set break as he took 82 percent of the points on serve overall.
He also finished off all four of his break opportunities.
Mayer defeated 11th-seeded Feliciano Lopez 7-6 (7-1), 6-2, winning two-thirds of the points in the second set.
That boosted Mayer into Friday's quarterfinals against top-seeded Roger Federer, who was a 6-4, 6-3 winner over Jan Hajek in his third-round match. Hajek did manage a break against Federer but Federer still took 70 percent of the points when serving.
No. 14-seeded Fernando Verdasco advanced when fourth-seeded Jerzy Janowicz retired with a right arm injury. Verdasco was up 7-5, 4-0 when the match was stopped. He is paired against Delbonis in the next round.
The other half of the bracket nearly played to form with three of the top four seeds around for Friday's play. The exception is 12th-seeded Fabio Fognini, who gained his spot in the quarterfinals with a 6-2, 6-4 win over Marcel Granollers.
The other quarterfinal sends third-seeded Nicolas Almagro against No. 5 Juan Monaco. Almagro was a 6-2, 6-3 winner over Guillermo Garcia-Lopez and Monaco defeated ninth-seeded Benoit Paire 6-3, 2-6, 6-2.
The formal name of the tournament is the best-at-home Open-German Tennis Championships.
Celebrity Families of 2014 [PHOTOS]