Stephens reached the fourth round by beating Mathilde Johansson of France 6-3, 6-2. Of the eight teenagers in this year's draw, the 19-year-old Stephens was the only one to even reach the third round.
"I'm excited because now I'm going to have more Twitter followers," Stephens said.
Sharapova has won three major titles in her career, but she still needs to win at Roland Garros to complete a career Grand Slam.
The 13th-seeded Ivanovic, a former top-ranked player who won the French Open title in 2008, committed 37 of her 40 unforced errors in the final two sets. Errani had only 18.
"In the third set I was creating a lot of opportunities and missing a lot of easy, easy finishing balls," Ivanovic said. "That's something that I'm not really happy about."
She didn't look anything like the player who overwhelmed seven-time Grand Slam champion Williams in straight sets on Wednesday.
Radwanska has been having a stellar year on tour, winning three titles and moving up to a career-high No. 3 ranking last month. But she is the only player in the top 10 that has never reached a Grand Slam semifinal.
Radwanska dropped to 3-10 against the Russian, including her only two losses in tournament finals.
"It's always disappointing when losing in the first week of a Grand Slam, that's for sure," Radwanska said. "Today I didn't play bad, she was just playing unbelievable."
Before Friday's match, Radwanska had been 38-7 in 2012, with six of those losses coming against top-ranked Victoria Azarenka.
Kuznetsova, who also won the U.S. Open in 2004, has struggled since winning the title at Roland Garros three years ago, only reaching one Grand Slam quarterfinal.
Rafael Nadal has made a habit of celebrating at Roland Garros, and it's not just because his June 3 birthday falls during the French Open.
This year, he is so focused on winning a record seventh French Open that he couldn't even remember exactly when he will turn 26.
"When is my birthday?" Nadal said Thursday after being asked if he had anything special planned. "I don't know the day."
When you play like Nadal on clay, the only date to remember is the day of the French Open final.
This year, that's exactly a week after his birthday.
"Difficult to celebrate ... when you are in the middle of the tournament, but sure, I'm going to go for dinner with the team," Nadal said. "I don't know if some family going to come. I don't know yet."
The second-seeded Nadal improved his French Open record to 47-1 on Thursday after beating Denis Istomin of Uzbekistan 6-2, 6-2, 6-0. If he wins five more matches, he will break the record of six French Open titles he shares with Bjorn Borg.
Nadal wasn't the only record-breaker on court Thursday. Isner, who won the longest match in tennis history two years ago at Wimbledon, lost 18-16 in the fifth set of another marathon match.