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.
Australia's Ellesse Andrew beat Hinze to the bronze medal.
Britain's women's track cyclists have dominated in the endurance events in the past decade but they have not had a world champion in the sprint since Becky James in 2013.
Finucane had been beaten to the gold in the team sprint by Hinze's Germany alongside Sophie Capewell and Lauren Bell.
But she dominated the individual event and her display means she will be one of the favourites at next year's 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.
