The world number 47 produced an unlikely comeback to lift the trophy, losing the first set and going 3-0 down in the second before powering to a 4-6 6-4 6-3 victory.
Ostapenko is also the first unseeded player to win the women's singles title at Roland Garros in the Open era and the youngest winner of the French Open for two decades, aged just 20.
The turning point in the final came in the second set when Ostapenko overturned a 3-0 deficit by winning four games in a row before claiming the deciding game with a forehand straight down the line.
The final set was a hard-fought affair which Halep was guilty of throwing away, having gone a break up to lead 3-1, but there was an element of fortune about Ostapenko's penultimate point which came when a backhand clipped the top of the net and bounced down on Halep's side to hand the Latvian a decisive break point.
She wrapped up the match in one hour and 59 minutes to make tennis history.