The 23-year-old Swiatek, back at Roland Garros where she captured a fourth French Open title last month, had the advantage of playing under the roof of Court Philippe Chatrier.
After two breaks in a comfortable first set, the top seed had to retrieve a break in the second set to get to 5-5 before sweeping the next two games and secured victory when Begu double-faulted.
Polish star Swiatek, who only made the second round on her Olympics debut in Tokyo three years ago, will face either Diane Parry of France or Argentina's Nadia Podoroska for a place in the last 16.
As the top seed enjoyed a winning start, play on the outside courts was delayed by three hours as rain swept across the French capital.