Carreno Busta won the opener 6-4 but could not build on the advantage and Djokovic came back to take the next three sets 6-2, 6-3, 6-4.
"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."
Djokovic required treatment on his upper left arm after losing the opening set and looked nervous against an opponent who was the beneficiary of his disqualification at the US Open in September.
But unlike in New York, Djokovic stayed within the rules to set up a semi-final clash on Friday with the fourth seed Stefanos Tsitsipas.
In the first of the men's quarter-finals in the top half of the draw, the 22-year-old Greek downed the 13th seed Andrey Rublev 7-5, 6-2, 6-3.
