Delbonis, 22, prevailed against the four-time tournament champion 7-6 (9-7), 7-6 (7-4) and will face 12th-seeded Fabio Fognini Sunday.
Fognini, who won his first ATP championship in Stuttgart last week, beat third-seeded Nicolas Almagro of Spain 6-4, 7-6 (7-1) in Saturday's other semifinal match.
Delbonis, a left-hander, seemed to catch Federer off his game.
Federer lost an early lead in the first set and committed consecutive groundstroke errors in the second set to give Delbonis a match-point opportunity.
"I feel like I'm dreaming," said Delbonis. "My key to the match was to enjoy playing this kind of player, because he's the best of all time."