She missed one match point as the German hung in there, but converted her second to become the youngest finalist at a WTA event since Nicole Vaidisova, the 2004 Tashkent winner.
"It's crazy," he said after the match.
"I thought I was out of the tournament in the qualifying and now I'm here. This is definitely the most drop shots I've hit in the tournament, it's normally not my most comfortable shot but it's been working."
'It's crazy!' - Gauff can't believe latest win at WTA Linz
Gauff is also in the doubles draw alongside her familiar partner Caty McNally.