The Australian served exceptionally well, sending down 25 aces and facing just one break point to advance in one hour, 47 minutes.
Djokovic put up a nice vengeance win against Murray in Dubai to start his year.
Kyrgios will be seeking to reach his first final of the season when he faces big-serving American Sam Querrey in the semis.
"I wasn't expecting an easy match, that's for sure", Djokovic said.
Serbian tennis player Novak Djokovic celebrates his victory over Slovak tennis player Martin Klizan. Djokovic was seeded number one and played some fine tennis heading into the last-eight encounter.
He fired a dozen aces, and broke Coric three times in the match - including twice in the final set to set up a clash with American Steve Johnson, a 7-6 (7/5), 6-3 victor over American wildcard Ernesto Escobedo.
"I wasn't thinking about winning at any stage", said Kyrgios. He bowed out of the 2017 Australian Open in the second round to Denis Istomin of Uzbekistan.
Kyrgios is only the second man - alongside fellow Aussie Lleyton Hewitt - to beat Novak Djokovic, Rafael Nadal and Roger Federer in his first meeting against each one of them.
Nadal, with yesterday's win took his tournament record to 13-0.
Another player who is struggling this year is Australian Bernard Tomic, who retired in the first set of his match against American Donald Young.
In the Brasil Open in Sao Paulo, second seed Albert Ramos-Vinolas advanced to the quarter-finals with a 6-4 7-6 (7/5) win over Gastao Elias.
He turned up on Tuesday and took little time in brushing off Japan's Yoshihito Nishioka 7-6 (7-2) 6-3 in a last-eight encounter.