Thiem cruised to the next round with a 6-4, 6-4, 6-2 victory over Gael Monfils. Anisimova became the youngest woman to reach the French Open quarterfinals in 13 years with a 6-3, 6-0 victory over Aliona Bolsova.
"It was a really good match," Thiem said. "My best match so far in this tournament."
Thiem will play Karen Khacanov in the next round. Khacanov outlasted Juan Martin Del Porto for a 7-5, 6-3, 3-6, 6-3 win in four sets.
"I'm just so happy I get to keep playing in front of the French crowd," Anisimova said. "It's amazing."
Anisimova will face defending champion Simona Halep in the quarterfinals. It only took 45 minutes for Halep to advance with a 6-1, 6-0 win over 18-year-old Iga Swiatek.