The 23-year-old Svitolina, who trains at Roland Garros and was a quarter-finalist in 2015 and 2017, had been tipped as one of the potential winners for the second year in a row after winning the Italian Open, but she never found her groove on Court One.
She dropped serve five times and made 29 unforced errors, bowing out on the first match point when she buried another forehand into the net.
Buzarnescu, who reached the final in Prague on clay last month, next faces American 13th seed Madison Keys, the 2017 U.S. Open runner-up.
Svitolina played down the expectations that were wighing on her before the tournament.
"Of course I believed that I can play well in the Grand Slams, but I didn't really think that I was a top favourite in this tournament," she told reporters.
"I don't want to think this way. But there are a lot of girls who are playing more consistent -- played more consistently than me on clay than I did."
Svitolina also said that it was hard to study her opponent, a late-blooming left-hander who has been rising fast in the rankings recently, breaking into the top 50 this season.
"Definitely it's a challenge, but we have to deal with those kind of things," she said.
"I have to really deal with it. It's tough. But I had to be ready, and today my game was not there."