The fifth seed, who last year became the first Italian woman to win a Grand Slam title, suffered an early break before securing four successive games.
She needed only 62 minutes to beat the 19-year-old and plays Anne Keothavong's conqueror Vesna Dolonts in round two.
Third seed Vera Zvonareva also advanced comfortably, defeating Lourdes Dominguez Lino of Spain 6-3 6-3.
The 26-year-old Russian, who lost in the final at both Wimbledon and the US Open last year, was guilty of 29 unforced errors compared to her 30-year-old opponent's 19 but took the first set in 38 minutes and secured the match in an hour and 21 minutes.
Denmark's top seed Caroline Wozniacki takes on Japanese 40-year-old Kimiko Date-Krumm later on Monday.
It was an impressive start at Roland Garros for Schiavone, who racked up 25 winners to Oudin's six on Court Philippe Chatrier.
Oudin, who burst onto the tennis scene two years ago with a fourth-round spot at Wimbledon and made the US Open quarter-finals, had no answer to the strength and agility of Schiavone.
"I pretty much got a clay-court lesson," the American admitted. "Practising this past week, I haven't played anyone that hits the ball that heavy.
"She is serving well, she's moving well, she pretty much doesn't have a weakness on the clay. I can see how she won the French Open last year.
"She's so confident here. You could see the shots she was going for and making them. I thought she played really, really well. There really wasn't much I could do."
The defending champion said: "I'm still shaking a little bit. I felt really happy to be there. The court is perfect. It is like when you go home and your mum does everything for you and you feel comfortable. I felt like that but with a lot of adrenaline."
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