Third seed Jelena Jankovic of Serbia was also beaten, going down 6-3, 6-3 against American Madison Keys, and second seed Petra Kvitova also struggled before beating fellow Czech Lucie Safarova 6-1, 5-7, 7-6 (4), winning the last five points to claim victory on her third match point.
Giorgi had previously demonstrated her grass-court skills by reaching the fourth round at Wimbledon as a qualifier in 2012, and she was a third-round finisher last year.
Azarenka won the first set from 4-2 down. Giorgi took a 5-0 lead in the second, saving five break points at 4-0. In the final set, Giorgi failed to serve out the match at 5-4, but broke again at 6-5 when Azarenka netted a backhand, and closed out the match at her second opportunity.
Despite her defeat, Azarenka believed she was almost ready to compete again at the highest level.
"I was really happy that I was there for every single moment, for every single ball," she said. "I felt that my level of concentration was really high. I just have to get into that rhythm."
In other matches, former champion Ekaterina Makarova defeated Italian qualifier Francesca Schiavone 7-5, 6-3, and seventh-seeded Italian Sara Errani was beaten 7-6 (5), 6-2 by American qualifier Lauren Davis.
There were also wins for sixth seed Flavia Pennetta, Slovak Daniela Hantuchova, Britain's Johanna Konta and American Varvara Lepchenko.
In the ATP event, seventh seed Santiago Giraldo of Colombia was beaten 6-3, 6-7 (5), 6-3 by Jarkko Nieminen of Finland.