The Romanian is into the last 16 of a major for the first time after hitting 31 winners to the Ukranian's 11, closing out the fourth seed in one hour and 37 minutes.
Buzarnescu claims efficient opening set
Buzarnescu got off to a quick start, breaking Svitolina in the third game with a scorching backhand winner down the line. The Romanian then held at love, an ace on game point giving her a 3-1 advantage.
The world number four was not without her chances and she was given the opportunity to level the set, holding break point in the sixth game, but failed to get a return into play, and Buzarnescu used her forehand to great effect, holding onto her single-break edge.
The 31st seed was now under consistent pressure on serve, but she came through with an ace on game point and a hold for a 5-3 lead.
Svitolina served to stay in the set, but the Ukrainian dumped a backhand into the net to give Buzarnescu a set point and a deep service return by the Romanian forced another netted reply, closing out the opening set.
Romanian claims gigantic upset in straight sets
Svitolina got off to a fast start in the second set, breaking in the third game for a 2-1 lead.
The resilient Buzarnescu broke right back, her biggest weapon, her forehand extracted another error from Svitolina on break point, levelling the set at 2-2.
Buzarnescu had gained the advantage again, but the first sign of nerves had crept in and she handed the break back when she missed an easy overhead to give Svitolina a 5-4 lead.
At deuce in the tenth game, the Romanian this time nailed an overhead to bring up a second break point of the game.
The set was even again after another Svitolina forehand miscue.
The 31st seed held with ease, bringing her to within a game of a career-defining win. Svitolina wasted a 40-15 lead as Buzarnescu's volley winner brought the game to deuce and the Romanian was at match point after a slice backhand forced yet another error.
A final backhand miscue from the Ukranian and the monumental upset was complete.
Buzarnescu addresses the media
Following the match, Buzarnescu said: "I only wanted to be aggressive and wait for the good shots and just go for it, play angles and move on, move on in the court."
As she was closing in on the biggest win of her career, she felt: "When you start seeing that you're kind of winning and leading in the match, thoughts are coming. So I had some ups and downs, but I really managed to get through it and play a good match today."
The Romanian had never won a match in the main draw of a Grand Slam event before this year's Roland Garros.
Now, she faces 13th seed Madison Keys in the fourth round.
"She's a really great player with an aggressive game and good shots," Buzarnescu said of the American.
"But, you know, I'm just going to play my own game and just go on the court as a normal match against a player like me."