Dervisoglu's Galatasaray teammate Kerem Akturkoglu and Atalanta defender Merih Demiral were also on target for the home team, along with Fenerbahce forward Serdar Dursun and Sassuolo defender Mert Muldur.
Turkey are second in Group G on 18 points, a point behind the Netherlands who can secure automatic qualification later on Saturday with victory over Montenegro.
Norway, who also have 18 points, were held to a 0-0 draw at home against Latvia.
Gibraltar were reduced to 10 men in the 23rd minute, when defender Jayce Olivero was sent off after a reckless challenge on Turkey fullback Zeki Celik.
Gibraltar are still searching for their first-ever points in World Cup qualification, having lost all of their 19 outings to date.