Rafa Nadal also reached the last four with a hard-fought 7-6(2) 6-3 victory over Japan's Yoshihito Nishioka and will next meet former U.S. Open champion Marin Cilic, whose quarter-final opponent Steve Johnson withdrew with an ankle injury.
The hugely talented Kyrgios's 2016 season was curtailed by an ATP ban for "not trying" at last year's Shanghai Masters but there was no doubting his commitment in his first meeting with 12-times grand slam champion Djokovic.
The 21-year-old held his nerve when the Serbian battled back for parity in the first set tiebreak, going 1-0 up when Djokovic fired a forehand long, and kept his concentration throughout an equally tight second set.
That also looked like going to a tiebreaker until Kyrgios pounced when leading 6-5 on serve, taking advantage of an uncharacteristically nervous service game from the former world number one to claim three match points.
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Kyrgios needed only one for victory, taking the initiative with a rocket of a return off the Serbian's second serve and finishing the match off with a fierce forehand winner into the backcourt.
Djokovic was returning to action as a wild card in Acapulco after a shoulder injury and playing only his second tour match since his stunning second round exit at the Australian Open at the hands of journeyman Denis Istomin.
Nadal, who won the title on both his previous appearances in Acapulco in 2005 and 2013 and reached the final at Melbourne Park in January, also had his back against the wall a couple of times against 21-year-old Nishioka.
The 14-times grand slam champion was 4-2 down in the first set and trailed 0-2 in the second but rallied each time and ended the one-hour, 57-minute match with an overhead smash.
"I did not play my best tennis, but that's why it has a merit," the Spaniard said. "I played great the previous rounds and today I was able to get over a tough situation. I'm happy to be in the semi-finals, that's great news."
(Reporting by Nick Mulvenney, editing by Amlan Chakraborty)
