On the comeback trail after taking a mental health break last year, world record holder Peaty qualified fastest for the final at the Aspire Dome with a time of 58.60 seconds but was half a second slower with the title on the line.
Veteran Fink, meanwhile, bravely fended off Peaty and Italian runner-up Nicolo Martinenghi to win in 58.57.
"I think I pushed it a little too far in the first 50 (metres) but at the end, it's good to be a part of (this)," said Peaty, who is aiming for a third successive gold in the event at the Paris Games.
"Definitely not an easy race to do... Of course, we've got a long way to go."
Diogo Ribeiro won the non-Olympic men's 50m butterfly title with a time of 22.97.
Germany's Angelina Kohler won the women's 100m butterfly title in a relatively weak field missing Canadian world champion Maggie MacNeil and the Fukuoka minor medallists, Zhang Yufei and Emma McKeon.
The 23-year-old Kohler burst into tears in the pool after winning in 56.28 seconds, 0.33 seconds ahead of American silver medallist Claire Curzan.
A day after electrifying the championships with the 100m freestyle world record in the men's 4x100m relay, China's Pan Zhanle failed to reach the semi-finals of the 200m, an event he was expected to contend for medals.
The 19-year-old never got out of second gear, posting a time of 1:51.03, more than six seconds off his personal best.
Pan's surprisingly anaemic performance may be with a view to the 100m freestyle for which he will be favourite. The heats start on Wednesday.
The women's 100m breaststroke world champion Ruta Meilutyte also disappointed, crashing out of the preliminaries.
Lithuanian Meilutyte, who took gold in the event at the 2012 London Olympics as a 15-year-old, was 17th quickest in the heats and more than three seconds off her championship swim in Fukuoka last July.
Meilutyte's setback followed Olympic champion Ahmed Fahnaoui's failure to make the men's 400m freestyle final on Sunday's opening day in a major shock.
World record holder Meilutyte will defend her 50m breaststroke title later in the meet.
China's Tang Qiantang topped qualifying for the 100m breaststroke final with a swim of 1:05.36.
With title-holder Katie Ledecky skipping Doha, Italy's former world champion Simona Quadarella is in the box seat for a second 1,500m gold.
The 2019 winner qualified fastest for the final with a time of 16:02.96. Fukuoka bronze medallist Li Bingjie was third quickest behind Germany's second-ranked Isabel Gose.
American Kate Douglass is set to defend her 200m individual medley title in the final later on Monday.
