Tuncay enjoyed a fine debut as Bolton overcame a stern test from League Two side Macclesfield.
The Silkmen took an early lead when Bolton's stand-in goalkeeper Adam Bogdan hesitated to allow striker Emile Sinclair to nip in and score.
Kevin Davies came on at the break but it was Tuncay who levelled when he volleyed in a Joe Riley cross.
Tuncay then sent Martin Petrov clear down the left and the Bulgarian drove in the winner.
Bolton had been made to wait for almost two weeks for a work permit to come through to allow 29-year-old Turkey international Tuncay to finalise his move from German side Wolfsburg.
He started up front alongside Robbie Blake and could have scored in the opening minutes but fired over from Petrov's cross.
And it was Macclesfield who took the lead in the 11th minute.
Tony Diagne's through-ball allowed Sinclair to sprint clear, round Bogdan, and slot into an empty net.
Blake, Darren Pratley, Chris Eagles and Zat Knight all went close before Tuncay equalised in the 56th minute with an acrobatic volley.
Extra-time was looming but Tuncay then released Petrov to race clear and fire in a late winner.