The world No 3 sealed his place in the French Open last 16 with a comfortable win over the Australian on??Court Suzanne Lenglen.
Murray ultimately made his experience count in a 6-4 6-2 6-3 victory but the Scot was not always in control of an enthralling contest.
Kyrgios kept the crowd entertained with a number of extravagant winners and acrobatic retrievals but the 20-year-old was too inconsistent to mount a sustained challenge.
“You go on the court expecting the unexpected,” said Murray, who will now play Frenchman Jeremy Chardy in the next round.
“You just have to be on your toes at all times and just try to be ready for something different really.
“In the matches I played against him, I feel like I have done a good job of weathering the storms during the matches.
“He always has some periods in matches where he’s on fire and hits some unbelievable shots but I just try to stay solid throughout and make it difficult.”
Murray has now won all three of his meetings against Kyrgios and after edging a close opening set, the British No 1 always looked likely to go through.
Kyrgios was given a warning for slamming a ball out of the stadium at the start of the final set and the Australian seemed to lose focus as he became increasingly distracted by an injury to his elbow.
“I have watched a lot of his matches, so obviously I know what his personality is like,” Murray said.
“He does have a game style and a personality on the court that lends itself to playing big matches on big courts.
“Sometimes you don’t feel like you’re able to control what’s happening out there because he’s hitting huge shots and sometimes playing shots that no-one else tries.
“You’re not expecting it so it’s obviously tricky to feel comfortable on the court and feel like you’re in a rhythm.
“That’s why he’s had so many good results, because he takes you out of your comfort zone a bit.”