The No 5 seeded 22-year-old held his nerve to post a 2-6 6-4 7-5 victory in Georgia and set up a clash with American Jack Sock.
Gojowczyk, the world No 108, raced through the opening set losing just five points on serve.
But Edmund, who saved a break point in the fifth game of the second, levelled matters by securing the only break of the set in game 10.
In the decider, Gojowczyk made the first break in game five only for the Yorkshire player to respond immediately.
The decisive break arrived the 12th game when Edmund clinched a hard-fought triumph – after two hours and five minutes – on his second match point.
Edmund admitted: “I was really pleased to get through that. He was hitting really cleanly in the first set and I wasn’t able to get into many points.
“I’ve been in Atlanta quite a while to get used to the conditions, so it’s nice to get wins under your belt and get some momentum for the rest of this hard-court swing.”
He now faces top-seeded Sock who disposed of Israel’s Dudi Sela 6-4 6-3.