Oudin defeated Coin 7-6 (3), 6-4 in just over two hours in the first reverse singles match.
The U.S. will next face either Russia or Serbia on April 24-25.
Story continues below ↓advertisement | your ad hereThe 18-year-old Oudin broke Coin in the fifth game of the second set and clinched the match when the Frenchwoman shanked a forehand.
Oudin had saved two break points at 5-5 in the first to force the tiebreaker, where she hit three straight winners to close out the set.
Alize Cornet had been scheduled to face Oudin but France captain Nicolas Escude decided to replace her with Coin after Cornet acknowledged she struggled with her nerves against Bethanie Mattek-Sands in the opening singles Saturday. Mattek-Sands rallied from 5-2 down in the first to win in straight sets, giving Cornet her sixth loss in six Fed Cup singles matches.
Both teams were missing their best players. Serena and Venus Williams skipped the event, while France played without Marion Bartoli, Aravane Rezai and Virginie Razzano.
The U.S. improved its record against France to 11-1 in Fed Cup play.
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