Murray, a four-time finalist in Melbourne and last year’s runner-up won 6-1, 6-2, 6-3 in two hours, seven minutes and will face Australian Sam Groth in the second round.
”Alex fought right through to the last point so he made it very competitive at the end in tough conditions today,” Murray said.
”It was very hot today so I was very glad to get it done in straight sets.”
Murray, who is looking to become the first man in the Open Era to win a Grand Slam title after losing the final four times, always had the contest under control.
He broke Zverev’s serve six times, saved six out of six break points and dominated when he came to the net, winning 12 points.
Eighteen-year-old Zverev, who works with Murray’s former trainer Jez Green, impressed with his court speed and coverage along with a big forehand.
But it was his inconsistency that was his major failing, getting himself into good positions but over-hitting his volleys.
Zverev suffered a nose bleed in only the second game of the match and needed a medical timeout before returning to the court.
The Scot broke Zverev twice on the way to taking the opening set in 35 minutes, and while the young German had Murray’s serve under pressure he never recovered from dropping his opening serve to lose the second set.
The final set went for almost an hour in the afternoon heat but Murray’s experience was key.
Murray has made it clear he will not hesitate to pull out of the tournament and dash home to be with wife Kim Sears for the birth of their first child.