Murray, who reached his fourth successive semi-final here despite a patchy performance in his 2-6, 6-1, 7-6, 6-1 victory over Kei Nishikori, admitted that he would need to play better against Wawrinka.
Murray beat the 21-year-old Khachanov 6-3, 6-4, 6-4 in two hours and four minutes to record the 650th win of his career. But once Ivan Lendl's pupil Murray settled into the match, he used his greater reach to stymie Nishikori's attempts to strike winners and managed to gut out the victory in breezy conditions.
He won the first set 6-1, led by two sets to one and looked poised to go a break up in the fourth when play was interrupted by a loud noise from the PA system.
"I'm very confident and hopefully we can have a great match".
"He's played fantastic this tournament so far", Murray said of the 32-year-old, the oldest Roland Garros semi-finalist since Jimmy Connors in 1985.
Nishikori, who advanced through a five-set thriller on Sunday, had to come back from one set down to see off Fernando Verdasco 0-6, 6-4, 6-4, 6-0.
The problem for Murray, who saw off eighth-seeded Kei Nishikori 2-6, 6-1, 7-6 (0), 6-1 on Wednesday, is that the man standing in his way is Stan Wawrinka, the 2015 champion.
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Djokovic was thumped 7-6, 6-3, 6-0 by the Austrian in the French Open quarter-finals. Each match I feel like I've played better.
Women's title favourite and 2014 runner-up Simona Halep trounced Spain's Carla Suarez-Navarro 6-1, 6-1 to make the last eight without dropping a set or breaking sweat.
"I am in semifinals with a very positive feelings", Nadal said.
The last French player in the tournament, 28th-seeded Caroline Garcia, is facing No. 2 Karolina Pliskova on Court Philippe Chatrier. You always have to expect guys to fight hard. "I think I'm playing better than a year ago". It's not what I'm playing for.
Svitolina beat Halep in their latest duel in Rome this year, but she nearly failed to set up another clash with the Romanian.
Nishikori surpassed Jiro Satoh for the most quarter-final appearances by a Japanese man, as he made the last eight for the second time in three years in Paris. "When I beat him in Rome, all was going well for me, everything was landing in.it will be a different story on Friday".