The world number two took the first two sets in 71 minutes but found the tall 18-year-old more of a handful in the third when he saved two match points before the Briton completed the victory in a little more than two hours.
Zverev, playing in the main draw at Melbourne Park for the first time, suffered a nosebleed while serving at 30-15 in the second game of the match on a sun-baked Margaret Court Arena, and was forced to call the trainer to staunch the flow.
Murray, who stayed on court and practiced his serve until Zverev returned, romped through the first two sets before the teenager produced some fight in a baseline battle in the third and made the Scot work for the win.
(Reporting by Greg Stutchbury; Editing by John O'Brien)
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