The third seed looked like he could be in trouble when he was pushed to a second-set tie-break after dropping the opener, but he won seven straight points and pulled away to win 4-6 7-6 (0) 6-2 6-4.
Djokovic said: “He was the better player for most of the first two sets. I was struggling to find my rhythm. I came into the match a bit sluggish, but I played a perfect tie-break and then played a couple of levels higher.
“It’s a big fight, it’s something that you expect, quarter-finals of a grand slam. You’re not going to have your victories handed over to you. I’m happy to overcome this big challenge.”
Djokovic struggled to deal with the power game of Russian Khachanov in the opening set and made some unexpected errors as he dropped serve in the fifth game.
Khachanov, who made the semi-finals at both the US Open and Australian Open, had another chance to break in the ninth game and, although Djokovic managed to hold this time, his opponent comfortably served out the set.
The momentum slowly began to switch in the second set as Djokovic, who was engaging in repeated dialogue with his support camp, began to apply some pressure to the Khachanov serve.
He did not manage to break but, not for the first time this fortnight, played a flawless tie-break and then benefited from a lucky netcord to win a long opening game of the third set on the Khachanov serve.
The only blip after that came with a loose game that saw him lose his break advantage at 4-3 in the fourth set, but Djokovic regained it straight away and served out the victory.
The Serbian is through to his 45th slam semi-final, one behind Roger Federer and seven adrift of Chris Evert’s all-time record, and now only two victories away from a record 23rd slam title.
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