The trio of 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 since the event was added to the Olympic programme at London in 2012, with the win capping an extraordinary day that saw the world record repeatedly lowered.
Britain first smashed China's old mark in qualifying, crossing the line in 45.472.
But it 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 a composed Finucane riding a storming last lap to make history.
"It didn't feel real all day, we just did every ride and 'oh faster, oh faster again," said Capewell.
Finucane is the new face of British cycling after bursting on the scene by winning the sprint world title last year in Glasgow.
"We nailed that final and I believed in us before we went up to do it but to execute lap one, lap two and lap three pretty much perfectly and then to cross the line first, I was like 'no way'," she said.
"It's just a dream come true and to have these two by my side is unreal."
Two-time defending champions China 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.
'Track so good'
Defending gold medallists and world champions the Netherlands were fastest through the men's team sprint qualifying in an Olympic record 41.279, just shy of their own world best.
Led by Harrie Lavreysen and Jeffrey Hoogland, the top two sprinters in the world, they came home ahead of Britain, the champions in 2008, 2012 and 2016.
The Dutch go head-to-head with Canada on Tuesday and Britain take on Germany as they bid to advance to a gold medal showdown.
Australia set the pace in the men's team pursuit qualifying with Oliver Bleddyn, Sam Welsford, Conor Leahy and Kelland O'Brien edging Britain.
Working together over 16 laps, they clocked 3:42.958 to go close to the world record Italy set when they won gold in Tokyo three years ago.
They will take on the fourth-placed Italians in round one on Tuesday as they continue their search for a first title in the event since Athens in 2004.
"The track is so good. It has been a bit of a surprise to see 42 seconds (3:42.958) to be honest," said Welsford.
"I was not expecting that. We had a lot more to give at the end of that race. We nailed the process and the preparation."
Britain own the most Olympic medals in men's team pursuit with 15, winning at Beijing, London and Rio before slumping to seventh in Tokyo.
After finishing second, they face third-placed Denmark, with the two winners of the next round clashing for the gold medal.
