The Black Cats laboured to a place in the fifth round of the competition, with Charalampos Mavrias's fifth-minute strike proving enough to down the Conference outfit.
He told reporters: "At this stage, it's about doing the job. We started well but after that we relaxed a little and it was hard.
"Credit to Andy [Thorn] and his team, they were organised and fought for everything. They were waiting for that one chance."
The North-East outfit discover who they will face in the next round of England's premier cup competition tomorrow.