The second seed marched into a 5-2 lead in the opening set but Vekic responded in inspired fashion to turn the match on its head with some stunning ground strokes.
Vekic saved two set points in a crucial opening set tiebreak and was the better player as she became the first Croatian to reach the Olympic women's singles quarter-finals since Iva Majoli in 1996.
US Open champion Gauff was left angry and in tears as an over-ruled line call saw her slip 4-2 down in the second set, the American being overheard saying "I'm being cheated."
Her long protestations with the umpire and supervisor came to nothing though.
The sun-baked Philippe Chatrier crowd got behind Gauff after that but ice-cool Vekic, seeded 13th, sealed the win as the American's game disintegrated. Olympic debutant Gauff will now focus on mixed doubles and women's doubles.
Gauff refused to blame the dispute for her loss but insisted, with some justification, that the point should have been replayed because the line judge's initial shout of "out" impeded her stroke.
"I felt that he called it before I hit and I don't think the referee disagreed but he just thought it didn't affect my swing which I felt like it did," she told reporters.
"I mean, there's been multiple times this year where that happened to me, where I feel like I always have to be an advocate for myself on the court.
"I feel like in tennis we should have a VR (video review) because these points are big deals. Afterwards, they apologise, but sorry doesn't help you once the match is over.
"But I'm not going to sit here and say one point affected the result today, I was already on the losing side of things."
Vekic suffered a heart-breaking loss to Jasmine Paolini in the Wimbledon semi-finals but has bounced back impressively in Paris.
She said the atmosphere on the court was one of the best she had played in.
"It was very, very tough. When we practised this morning the roof was closed so it took me a couple of games to adjust (to the heat). The ball was flying a little bit," she said.
"The support has been unbelievable and I'm having a great time in the village so hopefully I can win one or two matches more. Then we can talk about medals."