Lepchenko rallied past Roberta Vinci 2-6, 6-4, 7-5 after the Italian struggled with a leg cramp in the final set, which featured seven breaks of serve. She had treatment on her left thigh and rallied from 5-1 down in the third to tie it 5-5.
Then Lepchenko broke back and served out the match.
"I really lost my focus and got a bit distracted, which happens a lot to a lot of players," Lepchenko said. "I just kept on fighting, said to myself it's not the end of the match, you can still win it."
Lepchenko, who is originally from Uzbekistan, moved to the U.S. in 2001 and claimed political asylum. She officially became a U.S. citizen in 2011 and was part of the team at the London Olympics.
"I feel amazing to be part of this team, to have so much spiritual support behind me," Lepchenko said. "There's this feeling that I can't let my team down, so I try even harder, more than if I was playing for myself. The team is really important for me.
"I wasn't that nervous, because of my experience in the Olympics, where I was super nervous. I told myself I wasn't going to let emotions ruin my chances here. I was a bit stiff at the beginning, but then I got into it."
The seventh-ranked Errani needed just more than an hour to beat the 64th-ranked Hampton.
With Serena and Venus Williams and Sloane Stephens injured, Hampton and Lepchenko made their Fed Cup singles debuts. Veterans Francesca Schiavone and Flavia Pennetta, who have struggled lately, were left off Italy's squad.
Italy beat the U.S. in back-to-back Fed Cup finals in 2009 and 2010.
"We're going to try our hearts out tomorrow," U.S. captain Mary Joe Fernandez said. "I'm really proud of how (Lepchenko) approached the match and how disciplined she was. It's not easy the first time, especially when you're playing with a crowd from another country, and she did a good job blocking it out."
Italy will hope Vinci recovers for reverse singles on Sunday.
"I hope it's just a bit of cramp, but they're treating her," Italy captain Corrado Barazzutti said. "Roberta made a really great effort. She was injured, but she played the last few games with all of her heart."
Japan 1, Russia 1
The winner faces Serbia or Slovakia in the semifinals on April 20-21.
Czech Republic 2, Australia 0
The Czechs are one win away from the semifinals against Italy or the United States.
Stosur had a 4-1 lead and failed to serve it out at 5-3, despite three set points. She had 10 double faults in the set, which took 1 hour, 23 minutes.
The Australian failed to serve out the second set at 6-5, and Safarova converted her second match point.
In Sunday's reverse singles, Kvitova plays Stosur and Safarova takes on Gajdosova.
Serbia 1, Slovakia 1
NIS, Serbia -- Dominika Cibulkova of Slovakia retired with an injury, giving host Serbia a 1-1 tie in the first round Saturday.
Cibulkova led Vesna Dolonc 6-4, 5-4 when she developed a leg muscle strain, fell to the court and was taken off the court on a stretcher.
Information from The Associated Press was used in this report.