Berrettini struck down the tee serves at speeds of 130 and 133 miles per hour to power to set point.
Djokovic struck an inside out down the line forehand wide to end the set.
However after dropping set one, Djokovic cut his forehand errors and raised the level of his game to secure an early break in each of the following three sets.
Berrettini's level dropped as Djokovic worked to expose the Italian's weaknesses in his backhand and athleticism.
Serving for the match at 5-3 30-15, Djokovic struck an inside out down the line forehand winner before Berrettini netted a down the middle backhand return to end the match.
Djokovic, 26-0 in grand slams this year, will face the Olympic gold medalist and 4th seeded Alexander Zverev of Germany in a semifinal night match Friday at 4:00pm PST USA (7:00pm New York time).
Earlier this year Djokovic won the Australian Open, Roland Garros and Wimbledon and now only needs the US Open title to become the first tennis player since Steffi Graf of Germany in 1988 to win all four grand slams in one calendar year or the rare Grand Slam.
In 2015 the American Serena Williams found herself in the same position having won the first three grand slams of the year before losing in the semifinals of the US Open to the unseeded Italian Roberta Vinci in a shock upset.