The 44th-ranked Delbonis never faced a break point and broke the Russian twice in the first set to take control of the match.
Brown will play top-seeded Fabio Fognini of Italy in the second round.
In Monday's other matches, Alejandro Falla of Colombia defeated Jarkko Nieminen of Finland 6-2, 6-1, and Marinko Matosevic of Australia advanced when Michal Przysiezny of Poland retired when leading 6-2, 5-6.
- Sports & Recreation
- Nikolay Davydenko
- Federico Delbonis
- Ivo Karlovic