Robson, 19, managed to avoid following fellow Briton Heather Watson out at the first hurdle despite being taken to three sets by the 21-year-old Italian.
It was her first win since the Australian Open in January, where she lost to Sloane Stephens in the third round. She has lost in the opening round in three successive tournaments - to Daniela Hantuchova in Doha, to little-known Kazakh Yulia Putintseva in Dubai and to Sofia Arvidsson in Indian Wells.
Ranked a career-high of 43 in the world despite a 2013 win-loss record of 3-6 prior to the match, Robson took just 20 minutes to sweep to the opening set.
'Burnt out' Heather Watson will play on
20/03/2013 AT 20:49
Eighty-fifth-ranked Giorgi fought back to take the second stanza, breaking Robson - who committed a slew of double faults and errors - twice while being broken once herself.
However the 5'6" player slumped to a fourth loss from four matches this season in the decider against left-hander Robson, who had five inches on her and served six aces to the Italian's two overall.
It was their first meeting. Robson will now face number 32 seed Alize Cornet of France, who enjoyed a bye into the second round.
WTA Miami first-round results
Bethanie Mattek-Sands (U.S.) beat Monica Puig (Puerto Rico) 6-4 7-5
Francesca Schiavone (Italy) beat Jana Cepelova (Slovakia) 5-7 6-4 6-4
Christina Mchale (U.S.) beat Anett Kontaveit (Estonia) 6-2 6-1
Anabel Medina Garrigues (Spain) beat Alexandra Dulgheru (Romania) 5-7 6-4 7-6(3)
Donna Vekic (Croatia) beat Yulia Putintseva (Kazakhstan) 7-6(4) 6-0
Madison Keys (U.S.) beat Allie Kiick (U.S.) 6-0 6-0
Zheng Jie (China) beat Yaroslava Shvedova (Kazakhstan) 7-5 6-4
Silvia Soler Espinosa (Spain) beat Pauline Parmentier (France) 6-0 6-7(2) 6-2
Eugenie Bouchard (Canada) beat Shahar Peer (Israel) 4-6 6-1 6-4
Laura Robson (Britain) beat Camila Giorgi (Italy) 6-2 4-6 6-3
Victoria Duval (U.S.) beat Andrea Hlavackova (Czech Republic) 7-6(4) 6-4
Magdalena Rybarikova (Slovakia) beat Mallory Burdette (U.S.) 6-2 6-4
Simona Halep (Romania) beat Sabine Lisicki (Germany) 6-2 3-6 7-5
Karolina Pliskova (Czech Republic) beat Lucie Hradecka (Czech Republic) 6-2 6-1
Urszula Radwanska (Poland) beat Annika Beck (Germany) 6-3 6-1
Olga Govortsova (Belarus) beat Melinda Czink (Hungary) 3-6 6-3 6-2
Svetlana Kuznetsova (Russia) beat Lourdes Dominguez Lino (Spain) 6-3 6-3
Romina Oprandi (Switzerland) beat Arantxa Rus (Netherlands) 6-4 3-6 6-3
Garbine Muguruza Blanco (Spain) beat Katerina Siniakova (Czech Republic) 6-2 3-6 6-4
Daniela Hantuchova (Slovakia) beat Tsvetana Pironkova (Bulgaria) 6-2 6-4
Jamie Hampton (U.S.) beat Monica Niculescu (Romania) 6-4 6-3
Stefanie Voegele (Switzerland) beat Petra Martic (Croatia) 6-3 6-3
ATP Miami first-round results
Dmitry Tursunov (Russia) beat Tim Smyczek (U.S.) 7-6(4) 7-5
James Blake (U.S.) beat Ryan Harrison (U.S.) 6-2 6-2
Dudi Sela (Israel) beat Marinko Matosevic (Australia) 6-2 1-6 6-4
Nikolay Davydenko (Russia) beat Paolo Lorenzi (Italy) 6-0 2-6 6-0
Horacio Zeballos (Argentina) beat Thiemo de Bakker (Netherlands) 6-1 3-6 6-3
Jurgen Melzer (Austria) beat Ricardas Berankis (Lithuania) 3-6 6-3 7-6(1)
Lleyton Hewitt (Australia) beat Joao Sousa (Portugal) 6-1 7-6(3)
Albert Ramos (Spain) beat Jan Hajek (Czech Republic) 6-3 6-1
Igor Sijsling (Netherlands) beat Leonardo Mayer (Argentina) 3-2 (Mayer retired)
Grega Zemlja (Slovenia) beat Roberto Bautista (Spain) 3-6 0-0 (Bautista retired)
Xavier Malisse (Belgium) beat Rajeev Ram (U.S.) 6-4 6-7(6) 6-4
Tobias Kamke (Germany) beat Blaz Kavcic (Slovenia) 7-6(5) 6-3
Michael Llodra (France) beat Benoit Paire (France) 7-6(7) 6-2
Somdev Devvarman (India) beat Evgeny Donskoy (Russia) 4-6 7-6(5) 6-2
Victor Hanescu (Romania) beat Andrey Kuznetsov (Russia) 4-6 6-3 4-1 (Kuznetsov retired)
Lukas Rosol (Czech Republic) beat Gilles Muller (Luxembourg) 7-5 6-4
ATP Metz
'My tennis is getting better' - Murray feeling good as he sets up Moselle Open quarter-final
WTA Indian Wells
Osaka withdraws from Indian Wells