With Murray serving to stay in the match at 30 all, the Italian struck a powerful crosscourt forehand return.
Murray netted a crosscourt slice forehand to face matchpoint.
Berrettini netted a crosscourt backhand pass before Murray struck an ace down the tee to hold serve and level the set at five games all.
The two players entered a tiebreaker following a hold of serve from each.
Murray struck a down the line backhand passing shot winner to open the tiebreaker.
Berrettini hit four consecutive groundstroke errors to then trail 0-5.
Following several additional errors from the ground from both players, Murray led 8-3 in the ten point tiebreaker.
Murray framed two groundstrokes while under pressure from the Italian to reach 8-5.
Murray then serve and volleyed but hit a swing down the line forehand volley long to reach 8-6.
Berrettini struck a down the line backhand pass wide to face matchpoint at 9-6.
Murray hit an inside out crosscourt forehand return netcord winner to close out the match; his 40th winner of the day to 34 unforced errors.
Berrettini for the day hit 72 winners to 59 unforced errors including 31 aces to one doublefault.
In round two at the Australian Open on Thursday (Wednesday night PST USA), Murray faces either Thanasi Kokkinakis of Australia or Fabio Fognini of Italy.