Swiatek has spoken this year of wanting to win Olympic gold, boosted by the fact it is held on her favourite surface.
Four of her five Grand Slam titles have come on the Paris clay and she is a formidable frontrunner in a match.
However, she had to work hard against Begu, going a break down in the second set with her forehand and serve misfiring.
Begu served to force a decider at 5-4 but nerves played a part, with Swiatek breaking back and winning the match on a Begu double fault.
The match was played under the roof on Court Philippe-Chatrier, with the first day’s play heavily disrupted by rain.
No play has been possible on the outside courts, where Britons Jack Draper, Cameron Norrie, Dan Evans and Katie Boulter are due to begin their singles campaigns.
