The third seed beat an erratic and enigmatic Kyrgios 7-5 6-3 4-6 6-1 in under three hours to safely negotiate one of the perceived tougher round-one tests.
He put Murray under pressure from the start, and broke the world number three early in the first set. The Scotsman showed his class to break back immediately though, before breaking the Australian to take the first set.
© Getty Images
The 2012 champion began to find some consistency in his game in the second set, while Kyrgios became increasingly erratic. Murray broke early on, and held his serve convincingly to stamp his authority on the match.
The first five games went with serve before Murray earned two break points in the sixth, but he only needed one as Kyrgios's over-hit return saw the Scot win a crucial a game to move 4-2 ahead.
World number 37 Kyrgios spurned a number of chances to break straight back in the following game, but Murray edged him out en route to earning a two-set advantage.
The Aussie, who was yet to take even a set off Murray in any of their three previous meetings, threatened to end that hoodoo as he looked to spark a comeback by winning the first two games of the third set.
However, Murray hit straight back for what was the last break of serve until the 10th game, where Kyrgios threw himself a lifeline by breaking again to reduce the deficit.
His hard work proved in vain, though, as Murray established two breaks of serve en route to a 4-1 lead in the fourth set, before comfortably closing out the match from there.