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Djokovic, chasing a third French Open crown and record 23rd men's Grand Slam singles title, advanced to a 45th major semi-final and his 12th at Roland Garros.
He will take on world number one Alcaraz or Stefanos Tsitsipas, the player he beat from two sets down in the 2021 final, for a place in Sunday's championship match.
Djokovic dropped his first set of the tournament but dominated the second-set tie-break against Khachanov before putting his foot down to secure a ninth win in 10 meetings with the Russian 11th seed.
"I think he was a better player for most of the first two sets," said Djokovic.
"I was struggling to find my rhythm. I came into the match quite slow but played a perfect tie-break and from that moment onwards played a couple levels higher.
"It's a big fight, something you expect in the quarter-finals. You're not going to have your victories handed to you, you have to earn them."
Djokovic improved his record at Roland Garros to 90-16 after denying Khachanov his spot in a third successive Grand Slam semi-final. He will return to number one if he wins the title in Paris.