Tuncay was not introduced as a substitute until the 55th minute at the Lyme Valley Stadium and he took little time to make an impact, scoring his first in the 66th minute and then again on 79 and 89.
Robert Huth and Ricardo Fuller had already put the Potters in command before the Turkey star's introduction, while Michael Tonge also got on the scoresheet in the 71st minute.
A second Stoke side took on Gwent County League side Albion Rovers in Newport on Saturday and they were even more successful, romping to a 9-0 win.
Dave Kitson scored twice, with James Beattie, Ben Marshall, Matthew Lund, Florent Cuvelier and Tom Soares also netting. The rout was completed by an own goal and a strike from former Potter Wayne Thomas, who is currently training with the club.