World number 47 Ostapenko is the first unseeded player to lift the Roland Garros title in the Open era and the first Latvian major champion in history.
The 20-year-old is the youngest French Open winner since Iva Majoli in 1997 and the first player to win a debut tour- level title at a Slam since Gustavo Kuerten in Paris that same year.
"I am really happy. I have no words. It was my dream," said Ostapenko, who will rocket to 12th in the rankings.
"I was just trying to stay aggressive. A couple of games, everything turned my way. I was 3-0 down but I was fighting for every point.
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"I still cannot believe it. It was my dream always. When I was a child I was watching. It's amazing to play here. I just enjoy it so much."
For Halep, bidding to join compatriots Ilie Nastase and Virginia Ruzici as major winners, it was a second final defeat in four years at Roland Garros.
"I am sick in the stomach with emotions but hopefully I can win this one in the future," said Halep.
"I want to congratulate Jelena. All the credit for what you have done. Enjoy it, be happy and keep it going because you're like a kid."
Ostapenko set the tone immediately on a raucous Court Philippe Chatrier, disguising any nerves with an array of bruising groundstrokes to break Halep in the opening game.
But the Romanian delivered a swift counter-punch, hitting back straight away on the Latvian's serve before holding with ease for 2-1.
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More crisp striking from Ostapenko nudged the 20-year-old ahead once more, but the errors flew off her racquet just as regularly and Halep pulled 4-3 in front.
Ostapenko's wildness betrayed her as she served to stay in the set at 4-5, netting a forehand to offer Halep set point and drilling long to gift the Romanian the set.
The Latvian's mood soured further when Halep escaped from 0-40 down to begin the second set, the third seed then breaking Ostapenko to compound her frustration.
Halep consolidated with a gritty hold for 3-0 before Ostapenko fought off a trio of break points to stall her rival's momentum and spark an incredible comeback.
The errors crept back into her game and Halep responded for 4-4, but Ostapenko promptly broke to love to leave herself in sight of a deciding set.
She conjured up three set points with a laser forehand down the line, forcing the match to a third set with another sweetly-struck winner.
Ostapenko continued her feast or famine style but watched two break points flash by as Halep withstood a sustained assault.
Halep broke for 3-1 when Ostapenko drilled long, but the latter retrieved it the next game with a thumping return.
Halep, who saved match point against Elina Svitolina in the quarter-finals, had no response this time and Ostapenko sealed a breathtaking triumph with another break by hammering a backhand down the line.
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