Speaking on court after his 6-3, 6-4, 6-4 win, Murray said he was grateful to tennis fans for turning up at major events in the wake of terrorist incidents in Europe.
"There was a terrible tragedy in London, also in Manchester only a few weeks ago," he said during his post-match interview. "Paris has also had some problems in the last few years.
"I'm sure everybody will join me in sharing their thoughts and prayers with everyone who has been affected by this. It's something that has affected large parts of Europe and all over the world.
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"Obviously we want things to keep getting better, and I appreciate people coming to the tennis, creating a great atmosphere. I am grateful that I'm able to perform in front of you."
Seven people were killed and dozens injured in London on Saturday night in a van and knife attack on London Bridge and Borough Market.
Last month, 22 people died when a suicide bomber struck at an Ariana Grande concert in Manchester. A concert in aid of those affected was held in the city on Sunday night.
"The match itself was probably the most routine and comfortable win that Murray has notched since February, when he lifted his one title of the season to date in Dubai," says Simon Briggs of the Daily Telegraph.
"He has improved with each of his four matches in Paris to date and today was comfortably his most dominant win of the clay-court season, even if Khachanov was the bigger hitter."
The win takes Murray into the 29th quarter final of his career – "having started the French Open harbouring significant doubts about his form, the world number one has played himself into familiar territory", says The Guardian.