The British number one got past the controversial Australian in four sets, 7-5, 6-3, 4-6, 6-1.
Kyrgios is known for being outspoken – and entertaining – on court.
Murray says it's sometimes difficult to keep his focus when he plays him.
Murray will now face Adrian Mannarino of France next.
He's joined by two other British players in the second round, as Johanna Konta beat Louisa Chirico, and Aljaz Bedene progressed after Ernests Gulbis retired injured in the third set.
Continuing her comeback from long-term injury, Laura Robson was beaten by Russian Elena Vesnina, and James Ward lost to Thomaz Bellucci.
