The 44th-ranked Spaniard broke the Austrian's first service game to win the opening set, and saved three break points while 2-1 down in the second before adding another break.
Melzer returned from a toe injury and played his first match since losing to Jo-Wilfried Tsonga in the first round at Wimbledon.
The sixth-seeded Dutchman, who won the event in 2011 and '12, dropped serve early in both sets and failed to break Lorenzi's serve.
The 81st-ranked Argentine won just five points on serve in the opener and trailed 6-0, 1-4 when the match was interrupted for an hour by rain.
Monaco will next play No. 7 Jarkko Nieminen of Finland, who beat Teymuraz Gabashvili of Russia 6-4, 6-4.
- Sports & Recreation
- Paolo Lorenzi
- Pablo Andujar
- Jurgen Melzer