Finucane beat two-time world champion Emma Hinze in the semi-final round and then proved too strong for Friedrich in the gold-medal ride, winning 2-0.
New Zealand's Ellesse Andrews beat Hinze to the bronze medal.
The final night of competition in the Chris Hoy Velodrome provided a big upset as Colombia's Kevin Quintero left the big favourites in his wake to win the men's keirin gold.
New Zealand's Aaron Gate won the men's points race from Spain's Albert Torres and Belgium's Fabio Van den Bossche.
Local favourite Katie Archibald's hopes of bringing the curtain down on the meeting with gold in the omnium fell apart in a noisy Chris Hoy Velodrome.
Archibald, whose partner Rab Wardell died in his sleep from a cardiac arrest a year ago, launched a brave early attack in the final points race but ended up fourth with American Jennifer Valente retaining her title ahead of Denmark's Amalie Dideriksen and Lotte Kopecky.
"I'm frustrated about how I went through the event as a whole, but it's nicer to finish on a race where I feel like I've given it my all," Archibald said.
The host nation ended on top of the medals table though with their best showing in terms of golds since 2016 with the Netherlands second and the United States third.
Britain's fifth gold was provided by the flying Finucane.
NEW FORCE
Britain's women's track cyclists have dominated endurance events in the past decade but they have been in the doldrums in the sprints with the Becky James the last to win the individual title in 2013.
But in Welsh rider Finucane they have unearthed a new force just a year out from the Paris Olympics.
"I get the feeling we're going to be talking about Emma Finucane for a very long time," Britain's former world champion Chris Boardman, commentating for the BBC, said.
Finucane was beaten to gold by Hinze's Germany in the team sprint alongside Sophie Capewell and Lauren Bell but dominated a high-class field in the individual event.
In the qualifiers on Monday she rode the fastest ever 200m flying lap by a woman at sea level and she continued that speed all the way to the gold medal.
"I like to focus on me and what I am doing because you can get carried away. I was in the zone and quite aggressive. It is a mental game which I quite enjoy," Finucane, who out-foxed riders with far greater experience, said.
"It is super exciting for next year. I feel we have the momentum now. We are coming for sure."
The 24-year-old Quintero served notice of his speed when he won bronze in keirin at last year's worlds.
The focus on Wednesday was very much on Dutch duo Harrie Lavreysen and Jeffrey Hoogland, both of whom were seeking to win a third gold medal of the championships.
Scottish rider Jack Carlin was also tipped for a medal but all three were caught up in their own personal tussle and Quintero lit the fuse halfway around the final lap to storm to victory with Australian Matthew Richardson second and Japan's Shinji Nakano taking the bronze.