The summer signing replaced Mark Wilson in the 63rd minute with United trailing 1-0, and he scored twice and set up another for Ryan Gauld to help his side scrape into the third round.
Mitch Megginson looked to have forced extra-time with a stunning strike to make it 2-2 with three minutes remaining, but Ciftci struck his second in the 89th minute to settle a pulsating tie.
McNamara said: "I'm delighted for Nadir, I thought he was excellent.
"I told him to get in behind them with his strength. I knew he would cause them problems.
"It was a good ball from Paul Paton and an excellent first touch and great finish. He was excellent for Gauldy's goal as well, good strength in their box and a good ball across.
"He is a fantastic talent, I think everyone can see that.
"It's down to myself and the staff to make sure we get the best out of him every time he plays and that he keeps improving.
"He is only 21 years old and you can see he's a real prospect."
Dumbarton boss Ian Murray was disappointed with his side's defence for letting the Dutch-born Turkish youth international in behind them.
But he added: "Overall we have got to be delighted with the way the players applied themselves and went head to head with a good side."