Djokovic beat Spanish 17th seed Carreno Busta 4-6, 6-2, 6-3, 6-4 to reach the last four of a major for the 38th time as he chases a second French Open and 18th Grand Slam title.
“It took all my strength and lots of patience, he’s a very good player,” said Djokovic.
“He doesn’t make a lot of errors so all the points were long. I started this match a bit badly with bad movement. After I found rhythm and confidence. It was important to stay focused on the key points.”
The Serb now sits two wins away from becoming the first man in half a century — and only the third in history — to win all four Slams twice.
Semi-final #10 awaits @DjokerNole…#RolandGarros https://t.co/LyBR9bGljN
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However, the world number one required treatment on his upper left arm after losing the opening set and looked vulnerable against an opponent who was the beneficiary of Djokovic’s disqualification at the US Open.
Carreno Busta twice had chances to break in the second set but could not capitalise, instead losing his own serve on two occasions.
Djokovic unleashed a primal roar having fended off two break points in the first game of the third set and appeared in control at 3-0 until Carreno Busta clawed back to 3-all.
But the top seed reclaimed the initiative with another break at 4-3 on his way to taking the third set.
Djokovic grew increasingly animated when he missed out a string of break points early in the fourth set, but he eventually landed the decisive blow in the seventh game when Carreno Busta pushed a forehand wide.
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Earlier, Tsitsipas made his second Grand Slam semi-final with a straight-sets win over 13th seed Andrey Rublev, avenging last month’s loss to the Russian in the Hamburg final.
The Greek fifth seed downed Rublev 7-5, 6-2, 6-3 to match his breakout run at the 2019 Australian Open.
“I’m expecting since a young age to potentially triumph at these Grand Slams. I’m happy that I’m able to be in a position where I am today,” said Tsitsipas. “I’m chasing something spectacular.”
“Despite not having a good start and being a break down, I remembered what a big fighter I am,” he added.
“It’s also about fighting and trying to find solutions in difficult moments. I managed to put my brain to work.”
Tsitsipas, 22, has now won 15 successive sets at the tournament having been two sets down to Jaume Munar in the opening round.
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However, that streak looked in peril when Rublev, also making his first French Open quarter-final appearance, served for the opening set at 5-4.
A loose game from the Russian 13th seed allowed Tsitsipas to recover and he rattled off four consecutive games to take the lead.
Tsitsipas, last year’s ATP Finals champion, dominated the second set to leave Rublev on the brink, profiting from a favourable net cord to break at 3-2 after carving the opening following a 28-shot rally.
Only Djokovic has won more than Rublev’s three titles in 2020, but the world number 15 was powerless to stop a supreme Tsitsipas who closed out victory inside two hours.
Tsitsipas, who was the first Greek to have made the quarter-finals in Paris, crunched 35 winners and made just 17 unforced errors.