Coming into the contest on the back of 16 wins from his last 18 matches on Tour, Bublik produced another assured display on the Austrian clay, beating Cazaux 6-4, 6-3 in just 86 minutes.
The 28-year-old won 84% of points behind his first serve, overpowering his opponent with 26 winners across the two sets.
It continues a remarkable resurgence for Bublik, who has already claimed silverware in Halle and Gstaad this summer, as well as reaching the last eight at Roland Garros.
"I can't believe I did it as it was probably one of the toughest weeks in my life because coming from Gstaad, playing here, different conditions and raining," Bublik said in his on-court interview.
"I am pleased I got through and thank you for supporting me throughout the tournament."
The Kazakh will be keen to carry his momentum into the American hard-court swing, as he looks to climb into the top 20 and surpass his career-high ranking of world number 17.