Kyrgios is considered one of the most talented players on the circuit but the 20-year-old was too inconsistent to trouble the British number one, who came through 6-4 6-2 6-3.
It means Murray has made it past the third round at Roland Garros for the sixth consecutive time, barring a year out injured in 2013, and he will now play David Goffin or Jeremy Chardy for a place in the quarter-finals.
“It was a tough match,” Murray said. “He was going for huge shots and making me play far behind the baseline.
“I was trying to make as many balls as possible, trying to use slice and higher balls to break up his rhythm. That worked well.
“He was struggling I think with his elbow a bit so he had to slow down his serve – that’s one of his biggest weapons so that helped me.”
The battle lines were drawn in the first game as Murray showed his touch with two brilliant drop shots before Kyrgios displayed his power with two hefty aces.
The Australian gifted Murray an early break in the third game, however, when he double-faulted but Murray matched the mistake, handing the world number 30 a break back with a double fault of his own.
Kyrgios was a box of tricks, entertaining the crowd with flamboyant winners and unlikely retrievals, but too often it was genius one moment, madness the next.
In the seventh game, a stunning lob through his legs had the crowd inside the stadium on their feet, but moments later a backhand in the net and a forehand wide gave Murray the decisive break he needed to serve out the set.
After 10 games of drop shots, lobs and double faults the contest began to flatten out and a brilliant backhand winner from Murray secured another break in the fifth game of the second set.
The umpire came down to check the mark but Kyrgios, always ready to please the crowd, knew it was good and waved him back.
The Australian should have hit back immediately in the next game when, after opening up two break points, a poor Murray drop shot gave him the whole court to find a winner but the youngster somehow ballooned it long.
Murray never looked back, sealing another break via a wild Kyrgios backhand and serving out the set with ease before his opponent called for treatment on his right arm.
Murray lost his first service game in the third set but he broke back immediately, prompting Kyrgios to slam the ball out the stadium in frustration, for which he was given a warning.
Thereafter, there was little to trouble Murray as Kyrgios produced the occasional winner to keep the crowd enthralled, but the errors were far more frequent and Murray was able to break again to seal a comfortable win in one hour and 59 minutes.