Kenin, a late replacement for Madison Keys, beat Timea Bacsinszky 6-3, 7-6 (4) to wrap up the match at Freeman Coliseum. The Americans advanced to the 2020 World Group draw, while Switzerland was relegated to World Group II.
Kenin, 20, rallied from 3-0 down in the second set to force a tiebreaker and win her first Fed Cup match.
"I'm just really happy, and I really left it all out there on the court," Kenin said. "Sloane just said, 'You know what? Just fight. Leave it all out on the court.'"
In the opening match Sunday, Stephens beat Viktorija Golubic 6-3, 6-2. On Saturday, Stephens beat Bacsinszky 6-4, 6-3, and Golubic topped Keys 6-2, 6-3.
"Just to get back in the World Group is important," Stephens said. "We want to be competing, obviously, for another Fed Cup title."
The Americans, who won the 2017 Fed Cup, improved to 9-0 against Switzerland overall.