The trio of Emma Finucane, Sophie Capewell and Katy Marchant surged around the Saint-Quentin-en-Yvelines velodrome in a new world best of 45.186sec to beat New Zealand, with Germany earning bronze.
It was the first gold for Britain in the event since it was added to the Olympic programme at London in 2012.
Their win capped an extraordinary day on which the world record was repeatedly lowered.
Britain first smashed China's old mark in qualifying, crossing the line in 45.472.
But the record fell again to Germany in round one and minutes later New Zealand toppled it once more before Britain reclaimed the mark when they clocked 45.338.
With the title at stake, Britain again blazed to a new world best in the final showdown against New Zealand, with Finucane riding a storming last lap to make history.
While the velodrome in Paris is the normal standardised length, it is wider than usual, offering more momentum to the riders, particularly sprinters, with many tipping world-record times ahead of the event.
Two-time defending champions China only finished sixth.
It is the first time at an Olympics that the women's sprint has featured three riders, matching the men, working together to post the fastest time over three laps (750 metres).
In previous Games, there were only two riders over 500 metres.
