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Williams, who reached the semi-finals in Melbourne earlier this year, has been stuck on 23 slam triumphs – one shy of Court's career haul – since winning the 2017 Australian Open.
But the seventh seed in Paris did just enough in a challenging opening outing on Monday, the first official night match in French Open history, overcoming Begu 7-6 (8-6) 6-2 in one hour and 44 minutes.
Begu proved a worthy match, winning four straight games and at one point holding two set points in the opener, but the world number eight's quality and experience ultimately came to the fore.
Having charged into a 5-3 lead, Williams found herself at 0-30 down when serving for the first set, and an overhit slice followed by another errant, awkward backhand gave Begu the break.
More unforced errors followed from Williams as her hopes of clinching the set with a return break were dashed, before the 39-year-old scooped long on her own serve to give Begu the advantage.
The American regained her composure to hit back, though, and force a tiebreak. Begu had two set points to play with at 6-4 up, but Williams rallied, winning the next four to take a hard-fought lead.
A superb drop shot teed Williams up for a break at the start of the second set, yet she tensed up again in game six.
Williams was applauding Begu for a supreme forehand down the line, but the former world number one was soon frustrated as the Romanian clawed back to deuce three times.
Begu's resolve finally crumbled, however, and Williams was left to serve out a win which sets up a second-round tie with Mihaela Buzarnescu.