Keys dropped the first set to Cornet and then trailed 5-3 in the second before she came back for a 6-7 (4), 7-6 (4), 6-3 victory in the second singles match. France's Caroline Garcia began the day with a 6-3, 6-2 victory over Sloane Stephens.
The United States is playing without the Williams sisters in the World Group playoff on an indoor hard court at Saint Louis University. Serena Williams is taking a break and U.S. captain Mary Joe Fernandez said this month that Venus signed up for a tournament in Malaysia, not realizing that was the same week as the Fed Cup matches.
Stephens is scheduled to take on Cornet and Keys will face Garcia in singles play Sunday. The doubles match pits the U.S. team of Lauren Davis and Christina McHale against France's Virginie Razzano and Claire Feuerstein.
The winning nation will enter the World Group in 2015 and compete for the Fed Cup title. The loser will drop down to World Group II.
Cornet struggled with a leg injury through the final half of the match, twice seeking medical attention. She also finished with 15 double faults. Her last one sealed the victory for Keys.
Garcia broke Stephens' serve twice in the first set and three times in the second. After Stephens hit her final shot long, the 51st-ranked Garcia briefly went to her knees before putting her arms out as if to fly to the net for the post-match handshake.
The 20-year-old Garcia won her first WTA title in the Claro Open Colsanitas on Sunday.
"I can't be too disappointed with myself when you play someone playing that well," said Stephens, who is ranked No. 17. "I mean, you kind of just got to accept that and move forward and fight hard for the next day."