Svitolina, 18, is the first teenager to win a WTA tournament since Timea Babos took the title in Mexico in February 2012. She collected $40,000 along with the Baku Cup championship.
Svitolina had advanced to her first international-level final with wins over second-seeded Donna Vekic and No. 4 Alexandra Cadantu before meeting Peer in the final. Svitolina, a semifinalist at a WTA tournament a week ago, has won eight of her last nine matches.
Svitolina had a pair of breaks in each set against Peer. She lost a game on serve in each set but her serving game was generally solid. She finished off the match with her fourth ace of the day.
Peer, who won $20,000 as the runner-up, fell to 5-4 in tournament finals. She was last in a title match in July 2011 and her most recent championship was in September 2009.