Oudin, of Marietta, Ga., defeated Pauline Parmentier 6-4, 6-4 after Mattek-Sands outlasted Alize Cornet 7-6 (7), 7-5 in the first match.
"Alize is a great player and I knew I was going to have to grind the match out," Mattek-Sands said. "I knew it wasn't going to be easy and I think that coming with that mindset really helped me. ... It was very tight, it could have gone either way."
The 140th-ranked Mattek-Sands rallied from 5-2 down in the first set and clinched the tiebreaker after an unforced error from Cornet. She converted her third match point when Cornet netted an approach shot.
"I had my chance but I didn't seize it," Cornet said. "I can only blame myself."
It was Cornet's sixth loss in six Fed Cup singles matches.
"Stress prevailed over everything," she said. "I was shaking almost throughout the first set."
In the second match, Oudin broke Parmentier in the fifth game and held serve to take the first set on a forehand error by Parmentier.
In the second set, Parmentier had five break points for a 4-2 lead but failed to convert any of them. Oudin didn't miss her chance at 4-4 and broke Parmentier to get the opportunity to serve for the match.
Oudin sent a backhand into the net but won when Parmentier's backhand sailed long.
"Pauline had quite a few chances," said France captain Nicolas Escude. "But in tennis, you have to win the key points."
Reverse singles and the final doubles match are scheduled for today in the best-of-five series.
Both teams were without their best players.
Top-ranked Serena and No. 5 Venus Williams skipped the event while France played without Marion Bartoli, Aravane Rezai and Virginie Razzano.
The U.S. has a 10-1 record against France, sustaining its only loss in their last meeting, which came in 2003.
ZAGREB INDOORS: In Zagreb, Croatia, defending champion Marin Cilic beat third-seeded Juergen Melzer 7-6 (5), 6-4 to reach the final.
He will play Michael Berrer, who beat Philipp Petzschner 7-6 (6), 7-5 to reach his first ATP Tour final.
SA OPEN: In Johannesburg, Feliciano Lopez and Stephane Robert advanced to the final after eliminating the two highest-seeded players.
Lopez upset top-seeded Gael Monfils 3-6, 6-1, 7-6 (1), while Robert ousted second-seeded David Ferrer 7-5, 6-4.