The 35-year-old Vinci, ranked 166th in the world, lost 6-4, 6-3 on Tuesday to Golubic, 10 years her junior and ranked 99 places higher, reports Efe news agency.
Britain's Katie Boulter fought back from a set down to defeat Canadian Carol Zhao 3-6, 7-6, 6-2.
The 21-year-old Boulter, ranked 208 and playing at Miami on a wild card, booked her place in the main draw.
Her qualification will soften the blow to British hopes after Heather Watson was eliminated in the first round by Brazil's Beatriz Haddad Maia.
Russia's Natalia Vikhlyantseva, ranked 88 in the world, saw off the USA's Nicole Gibbs, world number 177, in straight sets 7-6 (7-3), 6-4.
Canada's Eugenie Bouchard, a former Wimbledon finalist who was once ranked world number 5, lost 6-4, 6-3 to Sweden's world number 116 Rebecca Peterson.
Several high profile players in the qualifying and 1st rounds at Miami will take to the court starting Thursday.
Several, including Caroline Wozniacki, Elina Svitolina, Karolina Pliskova, Garbine Muguruza, Caroline Garcia, and Jelena Ostapenko, on Wednesday were participating in the tournament's Media Day.
Wozniacki, who is coming off the back of her maiden Grand Slam title after her win at the Australian Open in January, was confident that finally securing her first Major trophy would allow her to play more freely, although she said her preparation for Miami would be the same as with any tournament.
"I think once I go to the next Grand Slam -- the French Open -- I'll feel differently, I'll feel less pressure. I think that will be the biggest difference for me", Wozniacki said.
(This story has not been edited by Social News XYZ staff and is auto-generated from a syndicated feed.)
