The top-ranked Murray became the 15th man to reach 650 tour-level match wins.
Third seed and 2015 champion Stan Wawrinka too reached the quarter-finals defeating Gael Monfils 7-5, 7-6 (9/7), 6-2.
Wawrinka is through to the last eight at Roland Garros for the fourth time after ending French hopes of a first men's singles champion in Paris since 1983.
The Swiss goes on to face Croatian seventh seed Marin Cilic for a spot in the semi-finals.
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Murray had never before played the unseeded Russian. The 53rd-ranked Khachanov made 38 unforced errors, compared to only 14 for Murray.
Murray will next play eighth-seeded Kei Nishikori, who recovered to beat Fernando Verdasco 0-6, 6-4, 6-4, 6-0.
"Kei is dangerous off both wings," Murray said. "Moves well."
In four previous matches on clay, Halep had never been able to beat Carla Suarez Navarro. But the Romanian took care of that problem on Monday, routing Suarez Navarro 6-1, 6-1 on Court Philippe Chatrier.
"I was expecting a very tough game," said Halep, the 2014 French Open runner-up. "I managed to push her back." Svitolina rallied to beat Croatian qualifier Petra Martic 4-6, 6-3, 7-5 on Court Suzanne Lenglen.
The 290th-ranked Martic was leading 5-2 and 30-0 on Svitolina's serve in the deciding set. The Ukrainian managed to hold serve, however, which allowed Martic to serve for the match at 5-3.
"It was amazing tennis from Petra today and she was really striking the ball well," Svitolina said. "I was really happy to come back in the second set. Thanks to the crowd for cheering me on. They gave me the energy to fight."
Former champion Stan Wawrinka was to play later Monday. The third-seeded Swiss, the French Open champion in 2015, will face 15th-seeded Gael Monfils.
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