Murray, 32, faced a tough test against world number 13 Berrettini, who reached last month’s US Open semi-finals.
The Scot came back from a break down in each set to claim his biggest win since returning from hip surgery in January.
Murray played at a high level, showing the battling qualities which are such a feature of his game.
He broke 23-year-old Berrettini, who was serving for the first set, before going on to dominate the tie-break.
Berrettini went 2-0 up in the second set but Murray again pegged him back to level and take the match following another tie-break.
Overall, Murray got 69% of his first serves in, winning 75% of those points and he also saved five out of seven break points.