Williams, 36, and Sharapova, 31, will meet in the fourth round on Monday.
Another former victor Garbine Muguruza also enjoyed an emphatic victory as the third seed, one of the pre-tournament favourites, thrashed Sam Stosur 6-0 6-2 in just over an hour.
"I think what happens there should definitely stay there and not necessarily (be) talked about in a not-so-positive way in a book".
However, Williams found the breakthrough once more and soon set up a tie with Sharapova.
They have verbally clashed in the past, such as a 2013 public spat about their private lives.
Williams has won 23 Grand Slam titles, Sharapova five.
Sharapova is seeded for the first time at a Grand Slam since her suspension for using meldonium and is playing her first French Open since 2015 after being refused a wildcard by tournament organisers a year ago. They are the only active women with a career Grand Slam; they are two of only six in history to accomplish that.
As far as Williams is concerned, she does not reserve any special on-court treatment for Sharapova - no matter what the Russian divulged in her autobiography. "She's been playing. for over a year now".
It has been nearly 2-1/2 years since Serena Williams and Maria Sharapova last faced off across the net but when it comes to these two great on-court foes, absence does not make the heart grow fonder.
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While Sharapova has spoken in the past of her struggles against Williams, who she has beaten only twice on the WTA Tour, she is in confident mood after a fine start to her campaign in Paris.
Apart from Sharapova's indiscretions, there were other things in the book that irked Williams.
In it the Russian described how she heard Williams sobbing loudly in the locker room afterwards, and claimed that defeat was the reason behind the American's dominant record over her since.
"When I left the match, I was with my father at the time, I said, "he's going to win and he might win a lot of Grand Slams behind that, because he was incredible", remembered Gasquet.
"The book was a lot about me. I'm playing tougher opponents and I'm hanging in there", said Williams, whose trophy collection includes three Suzanne Lenglen Cups. She won 12 consecutive points on her serve in one stretch.
There were plenty of other results involving top names at the French Open on Saturday.
Former champion Garbine Muguruza reached the last 16 with a comfortable 6-0, 6-2 win over Australia's Sam Stosur. Mens winners included 10-time champion Rafael Nadal, No. 3 Marin Cilic, No. 5 Juan Martin del Potro, No. 6 Kevin Anderson and No. 9 John Isner.
Even if Williams vs. Sharapova has not been competitive, it's still compelling.
The 2016 champion at Roland Garros lost her serve just once to the Australian, who won the 2011 US Open and made the semifinals in Roland Garros in 2016, 2012 and 2009.
"It feels like this suit represents all the women that have been through a lot mentally, physically, with their body to come back and have confidence and to believe in themselves", Williams said after her first win in Paris, a straight set win over 70th ranked Kristyna Pliskova that represented her first victory in a major tournament since winning the 2017 Australian Open - while pregnant.
Against Goerges, the careless errors were largely absent.