"Terrible tragedy in London, also in Manchester only six or seven days ago. Paris has had its problems too in recent years", said Murray. I'm sure everybody would join me in sharing our thoughts and prayers with those affected by this. "The fourth, I played flawless tennis". "I am grateful that I'm able to perform in front of you".
Murray made only 14 unforced errors.
The Scot will now face Japan's Nishikori, who downed Fernando Verdasco 0-6 6-4 6-4 6-0, and will take an 8-2 head-to-head record into Wednesday's contest.
Khachanov had never played in the main draw at Roland Garros before last week but the 21-year-old Russian, the youngest man to play in the fourth round here for eight years, is one of the game's outstanding young talents.
Cilic was leading 6-3, 3-0 after just 48 minutes when Anderson called it quits.
Khachanov was hoping to become the first player to reach the quarter-finals on his Roland Garros debut since Rafael Nadal in 2005.
Nishikori reached his 7th Grand Slam quarterfinal with his win over Verdasco and has made the last eight in Paris for the second time. I think I played the best match here in Roland Garros against Carla, which always was a tough match.
Elina Svitolina of Ukraine awaits for Halep in the quarter-finals after she defeated Petra Martić of Croatia 4-6, 6-3, 7-5.
The two met in the Rome final on the eve of Roland Garros when Halep was beaten and suffered an ankle injury which threatened her appearance in Paris.
"I sill feel the ankle some times but I am able to run". 'I think that's one of the great things about tennis is that, I don't know exactly how many different countries are represented here in the men's and women's events, but it's a lot.
In a late evening match played Sunday, defending champion Novak Djokovic eased into the quarterfinals outclassing clay court specialist Alberto Ramos-Vinolas 7-6 (7-5), 6-1, 6-3.
Nishikori beat Murray in a dramatic five-setter 1-6, 6-4, 4-6, 6-1, 7-5 past year to reach the semifinals at Flushing Meadows.
Second-seeded Karolina Pliskova of the Czech Republic reached her first French Open quarterfinal by coming back to edge 97th-ranked Veronica Cepede Royg of Paraguay 2-6, 6-3, 6-4.
Having started the French Open harbouring significant doubts about his form, Murray has played himself into familiar territory.
Pliskova was the runner-up at last year's U.S. Open.
Monday's other last-16 clash saw an all-French battle between Alize Cornet and Garcia, who are barely on speaking terms. If she had won that point, she would have served for the victory. When Khachanov served at deuce Murray went to break point with a thumping forehand return victor and converted it with a rally typical of so many he has won over the years.
Venus Williams opens play at French Open on Day 4
In the victory, Mladenovic won 70 percent of her first serve points and 43 percent of her second serve points. Serena Williams is at the French Open , after all - but as a spectator, not a competitor.