Both players had taken out some big names to make the final - the unseeded Peer took out No.3 seed Chanelle Scheepers in the second round, the No.7-seeded Svitolina scored back-to-back upsets over No.2 seed Donna Vekic and No.4 seed Alexandra Cadantu in the quarters and semis, respectively.
This was Svitolina's day though, as her power game was just too sharp for the more solid Peer, and that power was summed up on match point - she blasted a big ace out wide on the ad side to end it.
"Thank you Shahar for a great final," Svitolina said in the trophy ceremony. "It was a very tough match today and I had a lot of tough matches throughout the week, but I worked really hard coming into the tournament, and it turned out well for me. Thanks to everyone and I'll see you all next year."
"I'd like to congratulate you Elina - you played great," Peer said. "I was really happy to be in the final and in this position to win a title, and although I didn't win today, this tournament was great and I had a great time in Baku. Thanks to everyone for everything, and I hope to be back here next year."
At 18 years, 10 months and 16 days, Svitolina holds a special distinction now - she is the first teenager to win a WTA title in 2013. She is actually the first teenager in 17 months to win a WTA title (the last one was Timea Babos at Monterrey in February 2012, aged 18 years, 9 months and 16 days).
Peer, a former World No.11 who is now down at No.149 in the world after a rough couple of years, was playing her first WTA final since College Park two years ago. She is now 5-4 lifetime in WTA finals.
"I'm really happy I achieved the finals," Peer said afterwards. "I've had a lot of tough moments the last few years and I did play well this week - I have to only take the positives from this tournament."
The doubles went to No.2 seeds Irina Buryachok and Oksana Kalashnikova, who beat Eleni Daniilidou and Aleksandra Krunic in the final, 46 76(3) 104. Buryachok won her second WTA doubles title (she won here last year with Valeria Solovyeva) while Kalashnikova won her first WTA title of any kind.