The British twice came from behind in normal time to draw 3-3 and then clinched a 2-0 shoot-out victory.
It was the Dutch team's first loss at an Olympics since 2004 -- after a run of 21 wins -- and shattered their bid to become the first team to win three successive gold medals.
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Goalkeeper Maddie Hinch played the starring role for the British team, pulling off four saves in the shoot-out.
Britain, playing in their first final, had only won two bronze medals at past Games, in 1992 and 2012.
But they were in front in the opening period against the Dutch when Lily Owsley scored from close range after Sophie Bray had dribbled past three defenders.
Kitty van Male levelled in the first minute of the second period, firing the ball into the top corner of the goal.
Maartje Paumen, bidding for her third gold, put her side in front with a penalty corner.
But the lead lasted only a minute when Crista Cullen shot past an off-balance Joyce Sombroek in the Dutch goal.
The faster, quick-breaking Dutch restored their lead for a second time through van Male after a well-worked penalty corner had caught the British defence cold.
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But with eight minutes left, Nicola White saved Britain with a close-range effort after a goal-mouth scramble.
In the shoot-out, both sides missed their first two attempts but Britain grabbed the lead when Bray was fouled as she took her turn.
Helen Richardson-Walsh converted the ensuing penalty stroke.
Margot van Geffen hit the post which paved the way for Hollie Webb to hit the gold medal winner.
Earlier Friday, Germany defeated New Zealand 2-1 to win the bronze medal.
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