Earlier Sunday, world No. 1 Serena Williams boosted the U.S. to a 2-1 lead when she posted a 6-2, 6-1 win over 54th-ranked Sofia Arvidsson.
U.S. captain Mary Joe Fernandez made a last minute substitution putting in Venus Williams for the 16th-ranked Stephens in the fourth match. Stephens lost the first match of the weekend to Arvidsson.
"The hardest thing was trying to decide when to play which one," Fernandez said. "At the end of the day, experience just goes a long way. Venus has been there so many times before.
"To have her in that clinching match for me was really important."
Fernandez also knew that Venus Williams was desperate to play.
"To be honest, I wanted to be out there," Venus Williams said. "I told Mary Joe yesterday I feel good, I feel positive.
"I pumped myself up and I wanted to be there for the team."
The final score was 3-2 after the U.S. defaulted the doubles match, giving the doubles point to Sweden, due to injury of three of four of the American players.
The victory guarantees the U.S. a spot in the coveted 2014 Fed Cup World Group, while Sweden will play 2014 World Group II action.
Venus Williams posted 11 double faults in the match with seven of them happening in the second set. In the final game, she had two double faults.
"Yeah, definitely relieved," Venus Williams said. "I needed it to be over in two. Mary Joe needed it to be over in two. She was having a heart attack on the sidelines.
"It just feels good."
Serena Williams also had some serving problems, but just in one service game -- the third game of the first set. Serving into the sun, Williams offered Arvidsson three break points in that game -- the only break point chances the Swede had in the match -- but Williams managed to hold on to the game.
"I felt OK out there but the sun was intense," Williams said. "Those games like that are really important because if she would've won that game it would've been a tougher score, a different match.
"It was important for me to stay focused and fight for everything."
In the second set, Arvidsson held serve once, in the sixth game.
Except for that third game, Williams played focused tennis throughout the 57-minutes match.
She posted 35 outright winners to only eight for Arvidsson. She served 10 aces to none for Arvidsson. She broke serve on four of six break point offerings from Arvidsson.
Venus Williams improved her Fed Cup record to 15-2 in singles. Serena Williams improved her Fed Cup record to 10-0 in singles.
In other Fed Cup action, Russia and Italy advanced to the World Group final.