The 20-year-old, ranked 47, had never been beyond the third round at a Grand Slam before nor won a senior title.
Ostapenko's win denied Halep not only a first Grand Slam title but what would have been a win that elevated her to the No. 1 ranking, bumping the struggling Angie Kerber who still hangs on to the top spot.
Fittingly she thumped a backhand victor past Halep to seal victory, before tossing her distinctive green headband into the crowd with a look of disbelief.
"I want to say congratulations to Jelena, enjoy it and keep it going", Halep told the crowd and Ostapenko afterwards. I'm just so happy. The impatience of youth not only showed up in Ostapenko's play but also, occasionally, in her demeanor.
With the momentum fully behind her, Ostapenko powered to the second set with two more breaks of Halep, despite her experienced opponent pulling one back herself.
Halep broke for 3-1 when Ostapenko drilled long, but the latter retrieved it the next game with a thumping return.
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I know how exciting that's going to be for us as a team, but their fans are going to be into it. "Definitely heard the chatter". James recorded an NBA Finals-record ninth triple-double with 31 points, 10 boards and 11 assists, while Love had 23 points.
Halep's gutsy hold for 4-3 elicited the first signs of frustrations from Ostapenko, whose mood was not improved as a pair of wayward forehands three games later relinquished the first set. "It was always my dream, when I was a child I was watching players here. I just enjoy it so much".
"I was missing but a couple of games were decisive and everything turned my way. Even in the third I was 3-1 down but I tried to fight for every point".
In the third set, each player held serve for the first three games until Halep broke Ostapenko in the fourth game to take a 3-1 lead. "I played well", said Halep, also a runner-up at Roland Garros to Maria Sharapova in 2014.
Finally in front in the match for the first time, Ostapenko might have blinked but she held serve brilliantly for 5-3.
"That was a devastating loss", said NBC commentator John McEnroe after Halep fell to 0-2 in French Open finals.
Jelena Ostapenko was making an excessive number of unforced errors and that drove her to losing the first set by 6-4 and trailing 0-3 in the second set.
She trailed by a set and 3-0 but launched a memorable comeback with a remarkable display of fearless shotmaking. Leading a set and a break, Halep was unable to keep it together, throwing her racquet in frustration and getting tight in her shot selection, and that's when things started to unravel. At 30-40 on the Halep serve, Ostapenko had match point and she rifled an unstoppable backhand down the line to clinch victory, leaping in the air and turning in delight in the direction of her two coaches, her mother and the former Spanish player, Anabel Medina, who had been on the edge of their courtside seats throughout.
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