Turkey entered the Group C1 contest unbeaten in six matches, winning five of those en route to earning promotion into the second tier of the competition.
Viljormur Davidsen and Joan Simun Edmundsson gave the host, who is ranked 125th in the world – 83 places below Turkey – a two-goal lead early in the second half.
Gurler pulled one back in the final minute, but the Faroe Islands held on to extend its own unbeaten run to four matches.
Reflecting on a chastening loss, Istanbul Basaksehir forward Gurler told reporters: "We are incredibly sad. There is no excuse for this.
"Maybe we didn't want it as much as they did. We didn't fight. We completed the first four matches with very good results."
Quoted by Turkish outlet Aspor, Gurler added: "Maybe we were too comfortable, but we can't make any excuses for this evening. In the dressing room, there's a silence as if you were at a funeral home."