The 20-year-old has competed twice in the Australian Open, but through a wildcard each time.
Buzarnescu, 37, was a useful first opponent for the 23rd-seed. Though a former top-20 player she has not played on tour since 2022, having taken a break and suffered injury, and is in the foothills of a return.
Gibson took just over an hour to complete her match, having had to move swiftly to court 8 as she was following fellow Perth player Astra Sharma, who made quick work of dismissing American Emina Bektas 6-0 6-0 in 36 minutes.
Daria Saville was the day's third Australian success, but she needed rather longer to see off Briton Heather Watson 7-6 (7-2) 6-2, the match taking six minutes shy of two hours despite the second set score.
While that trio joined Bernard Tomic, Jason Kubler and Maddison Inglis in the second round of three that qualifiers need to win to secure their places in the main draws, there were four losers on Tuesday, reducing to six the original 15 Australians who entered qualifying.
Lizette Cabrera lost to Thailand's Mananchaya Sawangkaew 6-4 6-2 while Storm Hunter went down to China's Shuai Zhang 6-3 6-2.
In men's qualifying Li Tu let slip a first-set advantage to lose to French wildcard Clement Tabur 3-6 6-3 7-5 and Omar Jasika battled hard, but was eventually overwhelmed by Colombia's experienced world No.115 Daniel Galan 6-3 4-6 6-1.
They joined the five Australians eliminated on Monday: James Mccabe, Alex Bolt, Arina Rodionova, Taylah Preston and Priscilla Hon.
A dozen Australians have already earned a place through their ranking with Destanee Aiava and Tristan Schoolkate joining them via wildcards.
Meanwhile, there were several surprises in men's qualifying.
Borna Coric, the top seed, went down to Parisian Titouan Droguet, 5-7 7-6 (11-9) 6-2; Adrian Mannarino lost to Luca van Assche 6-3 7-5 in an all-French clash; and American Christopher Eubanks was beaten by Austria's Jurij Rodionov.
But Chile's Cristian Garin, Briton Dan Evans, America's Michael Mmoh went through, along with the 2014 US Open champion Marin Cilic of Croatia.
In women's qualifying the 38-year-old 2012 finalist Sara Errani rolled back the years after a sticky start to defeat Germany's Jule Neimeier 0-6 7-6 (7-4) 6-2.
Elsewhere Maya Joint joined Ajla Tomljanovic in the second round in Morocco with a 7-5 6-2 win over Ana Konjuh, of Croatia. Joint, who, like Tomljanovic, already has a main draw place at Roland Garros, will next play American Katie Volynets.
