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."
Swim Week Miami Beach 2014 [PHOTOS]