Berrettini needed medical attention near the end of the second set after appearing to almost faint on court but he battled on.
The Italian recently returned to the tour after an injury-plagued 2023, reaching the final of a Challenger event in Phoenix.
He broke Murray's serve in the first game of the match and was in firm control of the opening set.
But he faded badly in the second and called a medical time-out when 5-2 down after walking off the court looking dizzy.
The weather in Miami was not particularly hot or humid and more pleasantly mild than has often been the case at the tournament in the past year.
The 27-year-old former world number six, put up a better fight in the third, but a solid-looking Murray saw the match out in two hours and 47 minutes.
Murray, who won the Miami Open in 2009 and 2013 will meet 29th-seeded Argentine Tomas Martin Etcheverry in the next round.
In the WTA event, American Sloane Stephens beat Angelique Kerber 6-2, 6-3 in a meeting of two former grand slam champions.
Stephens, celebrating her 31st birthday, was on top throughout against the German who had shown good form in Indian Wells where she reached the fourth round.
Stephens, the 2018 Miami champion, will play Romania's 19th-seeded Sorana Cirstea in the second round.
Later on Wednesday Naomi Osaka was due to face Italy's Elisabetta Cocciaretto.
