It was a brief reprieve as Khachanov's aggressive volleying tactics earned him a third set-point chance and this time the Russian took it to clinch the opener.
In women's play, 13th-seeded Madison Keys reached the quarterfinals for the first time at Roland Garros after beating No. 31 Mihaela Buzarnescu of Romania 6-1, 6-4 on Chatrier.
It is the seventh time the world No.35 has reached the French Open fourth round, but it is the only grand slam tournament where he is yet to make the last eight.
Clearly this was incidental contact, but Dzumhur ended up with the loss to the No. 2 seeded Zverev in five sets.
"I think that's the matchup most of the fans in Germany and Austria were hoping for when they saw the draw", he said.
But Zverev edged a tense fourth set in which he was issued a code violation for being coached from the stands by his father Alexander Sr.
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In a drama-filled match, the German led 4-2 in the fifth set only for his opponent to break back and then go 5-4 up. "It's the best, a five-set match". The match kicked off with two breaks of serve and they found the range on serve to bring the next six games home easily before Sascha earned three break points in game nine.
"I've been down 2-1 every single time but I'm unbelievably happy to be into the quarter-finals".
Thiem, a semi-finalist at Roland Garros for the past two years, has won eight of his 10 ATP titles on clay and dominated on his favoured surface.
Zverev has youth on his side, but he has been spending a long time on court and his body is showing it.
"I made the final of the U.S. Open but lost to my friend Sloane Stephens", Keys said. But Dzumhur felt it was within his reach and also ran out to catch it.
Keys next faces unseeded Yulia Putintseva of Kazakhstan, who beat 26th-seeded Czech Barbora Strycova 6-4, 6-3.
Later on Lenglen, Australian Open champion Caroline Wozniacki - seeded second - had a tough-looking fourth-round match against Daria Kasatkina.