World number 10 and ninth seed Raducanu was beaten 7-6 (0) 6-2 by the Italian world number 29, who was tournament winner in Montreal last year.
Raducanu lost in the Citi Open quarter-finals to Liudmila Samsonova in Washington last week and preparations for her US Open title defence later this month were dealt another blow in the city of her birth.
The British teenager squandered chances to seize the initiative in either set and let slip a 2-0 lead in the second before bowing out in one hour and 49 minutes.
Both players lost serve three times in a hard-fought opening set before Raducanu dug deep to fend off set point in the 12th game to force a tie-break.
But the 19-year-old Briton lost her way and failed to win a single point as Giorgi forced four mini-breaks to seize control and take the first set after an hour and 11 minutes.
Raducanu, who had failed to convert game point for a 5-3 lead in the opening set, broke Giorgi's first service game in the second to lead 2-0.
But Giorgi broke straight back and reeled off the next five games to set up a second-round clash with Belgium's Elise Mertens.
Earlier, 10th seed Coco Gauff cruised into the second round by beating fellow American Madison Brengle 6-1 6-3, while Australian Ajla Tomljanovic, Spaniard Sara Sorribes Tormo and China's Qinwen Zheng all progressed.