Oudin gave the hosts a perfect start when she recorded a 6-3 6-3 win over Alla Kudryavtseva before Dementieva leveled the tie at the indoor hard court in Birmingham, which kept the players away from the storms battering Alabama.
The Olympic champion, ranked sixth in the world, defeated Bethanie Mattek-Sands 6-4 6-3 set the stage for a tense final day that will begin with her reverse singles rubber against Oudin.
"It always feels a little bit more pressure when you have to win your match," Dementieva told a news conference. "I expect a tough match against Melanie.
"I think this kind of court suits her game.
"Physically she looks pretty fit. ...Tomorrow I will be aggressive from the beginning, try not to go to long rallies because I know she feels very comfortable at the baseline."
Oudin, shouldering the burden of being America's top-ranked singles player after Serena and Venus Williams declared themselves unavailable, recovered from slow starts in both sets to beat Kudryavtseva, ranked almost 50 places below her and making her Fed Cup debut.
"I thought I played a really good, solid match all the way through ... a little rocky in the beginning of the first set, but I thought I closed it out really well," Oudin said.
"I'm gonna go into tomorrow just like any other match I'd play. I've played Elena twice now. We've played really good matches. I played well both times."
The best-of-five tie will be decided on Sunday with two reverse singles and a concluding doubles rubber.
The winners will advance to the November final against either holders Italy or Czech Republic. Italy lead 2-0 after the opening day in Rome.
(Reporting by Julian Linden, editing by Greg Stutchbury)