The 37-year-old American was coming off a straight-set victory over younger sister Serena a night earlier, ending a three-match skid against her sibling.
"It was a quick turnaround from last night when it feels like, 'Hey, this is a final' to 'You're only in the fourth round,'" Williams said on court.
Playing under cloudy skies in 80-degree heat, Williams had her hands full with Sevastova, who used drop shots, top spin, lobs and jerked her 10-years-older opponent side to side.
"There were some points where she just played and it was too good," Williams said. "I thought I was in control of the point and winning the point, and she turned it around. It's just real talent."
After early losses in her first two tournaments of the year, Williams hasn't dropped a set at Indian Wells, where she has yet to reach the final in six previous appearances.
Awaiting Williams in the quarterfinals is 27th-seeded Carla Suarez Navarro of Spain, who beat American wild card Danielle Collins 6-2, 6-4.
Simona Halep of Romania is on track to retain her No. 1 world ranking after defeating Wang Qiang of China 7-5, 6-1 in the fourth round.
Halep is the only former Indian Wells winner left in the women's draw after No. 20 Daria Kasatkina upset No. 2 Caroline Wozniacki 6-4, 7-5. The Dane blew a 3-0 lead in the first set and was on the run most of the match before losing for the second time this year to the 20-year-old Russian.
"She outsmarted me," Wozniacki said.
By reaching the quarterfinals, Kasatkina is projected to surpass her career-best ranking of 17th. She has beaten all four current Grand Slam titleholders in the past year, including U.S. Open winner Sloane Stephens in straight sets this week.
"I'm playing best matches against the best players," Kasatkina said. "But as I say, if you want to be on the top, you have to beat the top players. So quite simple rule."
Halep improved to 17-1 this year and needs only to reach the final to stay at the top.
Anisimova, a 16-year-old wild card, won her first three WTA Tour matches at Indian Wells, beating Pauline Parmentier, No. 23 seed Anastasia Pavlyuchenkova and two-time Wimbledon winner Petra Kvitova without dropping a set.
"She's 16, so she's not scared," Pliskova said of Anisimova. "You can see she's hitting the ball without thinking. But everybody is playing like this when they are 16. I think this will change a little bit in the future, but for sure the game is good and not really any weakness. I think everything is pretty solid."
Halep attended Venus Williams' straight-set victory over 36-year-old sister and new mother Serena on Monday night.
"I love the way that they are motivated and they are still playing at this age, Serena with the kid," Halep said. "It's a great thing what they do for sport, and it's great that tennis has them. I have many things to learn from them. That's why I'm trying just to go in to watch every time I can."
Monfils used an eight-minute injury timeout in the second set to get his back worked on before returning to the court and getting broken in fourth game. He then walked to the net and ended the match.
Cilic won 19 of 23 points on his first serve but failed to convert four break points against 31st-seeded Kohlschreiber, who snapped a 12-match skid against top-10 opponents.
Herbert remains in contention for the $1 million bonus offered to a player who sweeps the singles and doubles titles.
With Cilic, Ferrer, Novak Djokovic, Monfils and John Isner all eliminated from his quarter of the draw, things are looking wide open for del Potro, the 2009 U.S. Open champion. Top-ranked Roger Federer of Switzerland and del Potro are the highest seeds remaining.
"I'm not thinking about that. I just want to keep winning," del Potro said. "Still far away from the final, but of course, any chance to play with Roger would be great."
No. 18 Sam Querrey rallied past Indian qualifier Yuki Bhambri 6-7 (7), 6-4, 6-4 and will play No. 28 Feliciano Lopez of Spain. Lopez spoiled a potential all-American quarterfinal by beating No. 8 Jack Sock 7-6 (8), 4-6, 6-4.