The second-seeded Sharapova handled Stephens' strong serve well, accumulating 12 break points and converting eight.
In 2012, Sharapova completed a career Grand Slam at Roland Garros, adding that trophy to ones from Wimbledon in 2004, the U.S. Open in 2006 and the Australian Open in 2008.
Sharapova has won all eight sets she's played so far this year and will face No. 18 Jelena Jankovic in the quarterfinals. Jankovic, the 2008 U.S. Open runner-up, advanced with a 6-0, 6-2 victory over 54th-ranked Jamie Hampton of the U.S.
"Today was just a tough day. Really was. I really don't know what else to say," said the 23-year-old Hampton, who surprised 2011 Wimbledon champion Petra Kvitova on Saturday to earn a debut appearance in the fourth round of a Grand Slam tournament. "Just everything that was hitting my racket was going out today, unfortunately. Everyone has those days. Got to learn to get through them."
Her match against Jankovic, the 2008 U.S. Open runner-up, was the last of the day and was played in a nearly empty Court Suzanne Lenglen, following a five-set men's match.
"That match lasted, like, five hours. I had coffee, as well," Jankovic said about waiting around for her turn. "I was trying to stay awake."
The 20-year-old Stephens, who upset Serena Williams en route to reaching the semifinals of the Australian Open in January, fell to 1-8 against women ranked in the top five.
Stephens lost in the French Open's fourth round for the second year in a row. Last time, at 19, she was the first U.S. teenager to make it that far in Paris since Williams did it in 2001.
During an on-court interview after Monday's match, Sharapova, who's 26, was asked about facing a member of her sport's "younger generation."
"I still like to consider myself young," Sharapova said with a smile. "Maybe not the 'young generation,' but somewhere in the middle."
With wind whipping around Court Philippe Chatrier -- some spectators huddled under blankets -- Sharapova managed to handle the conditions well.
At 3-all, she held three break points, and she took that game when a good return off a 103 mph serve forced Stephens into an error. That little lead was enough to take the opening set, which Sharapova served out at love.
Stephens earned three break points the entire match and took advantage of only one. They all came in the second set's second game, when a cross-court forehand winner by Stephens made it 1-all.
But Sharapova broke again to go ahead 3-2 when Stephens put a backhand into the net. From there, Stephens' body language was filled with slumping shoulders and a hanging head.
Seeded No. 3, Azarenka improved to 26-2 in 2013. She has won the Australian Open each of the past two years, and Roland Garros is the only major tournament where she has yet to reach the semifinals.
Play began in weather so cool both players dressed in long sleeves, and Azarenka wore leggings under her skirt. From the baseline the two women traded strokes and grunts.
"Ah-heee!" Azarenka shrieked.
"Wah-hooo!" Schiavone responded.
There were four consecutive service breaks before Azarenka took the lead for good, holding to make it 4-3. The superior pace and depth of her shots began to wear on the unseeded Schiavone, and Azarenka took charge by winning 11 of 12 points during a stretch late in the first set.
The Belarusian broke serve seven times and committed only 14 unforced errors.
"It was definitely a very good challenge for me to play against Francesca, especially knowing how well she's done here in the past," Azarenka said. "I'm glad with the way I played today."
"The next step is quarterfinals," Azarenka said. "Never been past it here, so it would definitely be a much better result."
The 12th-seeded Kirilenko lost in the fourth round at Roland Garros in 2010 and 2011.
She trailed 4-1 at the start against Mattek-Sands, who stunned 2011 French Open champion Li Na in the second round.
But Mattek-Sands' lead began slipping away, and while ahead 5-4, she asked for medical attention. From there, Kirilenko won four games in a row.
Kirilenko held on, though, and faces Azarenka next.
Information from The Associated Press was used in this report.