Chasing her 18th Grand Slam title, the top-seeded American overwhelmed the 94th-ranked South African in the opening match on Court 1 at the All England Club.
Williams served eight aces and had 26 winners. After dealing with back issues earlier this season, Williams says she's feeling better.
"I felt OK today," she said. "I just had a goal and trying to do better each day. ... You know, I don't want to feel great necessarily right now because hopefully I have several more matches I can play. But the goal for me is just to feel a little better every day."
The 30-year-old Scheepers, who had never won a match at Wimbledon until this week, had only five winners to go with 13 unforced errors.
Williams, coming off a second-round loss at the French Open, is playing in her 15th Wimbledon and 56th Grand Slam tournament.
Maria Sharapova, the 2004 champion and fifth seed, beat Timea Bacsinszky of Switzerland 6-2, 6-1. Other seeded winners included No. 9 Angelique Kerber, No. 13 Eugenie Bouchard and No. 19 Sabine Lisicki.
Williams next plays Cornet, who beat her earlier this year.
"Cornet, she does everything good," Williams said when asked about the upcoming match. "That's why she was able to get a good win off of me. She never gives up, she's a big fighter. She gets a lot of balls back.
"But I'm ready. It's time for me to step up my game and do what I know I can do and do what I'm best at."
The Associated Press was used in this report.