"She came out and started playing a lot better and wasn't missing anything when she came back out," Oudin said. "She changed her game a little bit. I did the best I could."
Reverse singles and doubles were scheduled for Sunday in the best-of-five series.
The U.S. has won all nine of its previous meetings with Italy. But the Americans are without Serena and Venus Williams, who opted not to play after meeting in the season-ending tour championships last weekend in Doha, Qatar.
Oudin, who reached the quarterfinals of the U.S. Open, faces Pennetta in the opening singles on Sunday.
The 11th-ranked Pennetta used her consistent baseline game to wear down the 132nd-ranked Glatch, breaking early in the first set to set the tone, then cruising in the second set.
The 6-foot Glatch likes to rely on her serve, but she won only won 43 percent of the points on her first serve.
"My serve kind of let me down," Glatch said. "I didn't win a lot free points off my serve, so it made it very difficult to hold serve."
Oudin took a 4-2 lead with an early break in the first set before the rain delay. Schiavone broke back in the first game when play resumed, using a drop-shot winner on her first break point.
With the crowd chanting "Fran-CES-ca, Fran-CES-ca" between points, the 18-year-old Oudin maintained her concentration and saved three break points to take a 5-4 lead. She flattened out her forehand for a couple of winners up the line and used an effective drop shot of her own on occasion, yelling "C'mon" whenever she won a big point.
"It's absolutely key playing at home," Schiavone said. "It really helps having the crowd behind you on the important points and the crucial points of the match."
In the tiebreaker, Schiavone's bigger serve and more powerful groundstrokes made the difference, and she landed an inside-out backhand return winner on the line on her first set point.
"I had lots of chances in the first set, so losing was pretty tough," Oudin said. "But I fought hard throughout the whole match."
Oudin broke in the opening game of the second set but Schiavone broke right back and took a 5-1 lead against the teenager from Marietta, Ga. Schiavone then dropped her serve, but broke in the next game to close it out when Oudin landed a forehand in the net.
"Fed Cup, you have to go a match at a time," U.S. captain Mary Joe Fernandez said. "We've seen a lot of comebacks from 2-0 down. It's a new day."
Glatch routed Pennetta 6-1, 6-1 in the first round of this year's French Open, but she was never in control on the slow clay during a cool and overcast day in southern Italy.
"She definitely played better than last time," Glatch said. "The conditions were also different. It was much quicker and hot that day and very dry, and I served extremely well. I was more aggressive in that match, and I probably should have tried to be in this match."
The 20-year-old Glatch broke Pennetta in the opening game of the second set, but the Italian took control.
"After I broke her, the next two games she really stepped it up," Glatch said. "She hit a couple great shots."
Pennetta finished with 33 winners to Glatch's 14 and broke the American five times. Glatch struggled with her mobility, watching several winners whizz by her within reach.
"She has a pretty varied game," Pennetta said. "She can slice the ball, hit it hard, topspin. You need to move her around because once she starts moving she has trouble."
One of the few things Pennetta struggled with were low balls, with the heavy clay hardly providing any bounce.
"There was a bit too much clay on the court, but that's what we asked for," Pennetta said.