Murray, who lost in last year's tournament final, was broken at the beginning of the final set before returning the favor on the ninth game en route to posting a 6-7 (3), 6-2, 6-2, 7-6 (3) victory at Paris.
"I expected it to be very tough," the 30-year-old Murray said of facing the 50th-ranked Klizan. "He goes for huge shots. He can hit forehand winners from anywhere on the court. Sometimes you think you've hit a good shot and he comes up with unbelievable power from very defensive positions and makes winners. It's very tough."
Murray of Scotland advances to face 29th-seeded Juan Martin del Potro of Argentina, who moved on after Spaniard Nicolas Almagro retired in the third set because of an injured left knee. The two traded 6-3 wins and were knotted at 1-all in the third set when Almagro fell to his back, prompting del Potro to climb over the net and check on his opponent.
"In my opinion Juan Martin plays better than what his ranking is," Murray said. "He's come back from injuries and he's had a lot of tough draws this year. I think he's played Novak (Djokovic) a couple of times. He's played a lot of top 10 players very early in the tournament. But he's playing well this year and it will be a tough match."
Third-seeded Swiss Stan Wawrinka, who was the 2015 French Open champion, hit 45 winners to best Ukrainian Alexandr Dolgopolov 6-4, 7-6 (5), 7-5.
Seventh-seeded Croatian Marin Cilic benefited from 11 aces to breeze to a 6-3, 6-2, 6-2 win over Konstantin Kravchuk, spoiling the Russian's debut at Roland Garros. The seventh-seeded Cilic advances to face Feliciano Lopez, who outlasted fellow Spaniard and 30th-seeded David Ferrer 7-5, 3-6, 7-5, 4-6, 6-4.
Eighth-seeded Kei Nishikori of Japan posted a 6-3, 6-0, 7-6 (5) victory over Frenchman Jeremy Chardy.
Frenchman Gael Monfils, who is seeded 15th, had little issue dispatching Brazil's Thiago Monteiro, recording a 6-1, 6-4, 6-1 triumph.
Khachanov collected 12 aces in the win and advances to face 21st-seeded American John Isner, who secured a 6-3, 7-6 (3), 7-6 (2) victory over Italian Paolo Lorenzi.
Fabio Fognini, the tournament's 28th seed, recorded a 6-4, 7-5, 6-3 win over fellow Italian Andreas Seppi.
Unseeded Hyeon Chung of South Korea and Kyle Edmund of Great Britain recorded straight-set victories on Thursday.