World number seven Kerber fought back from dropping the opening set to beat former US Open champion Stosur 4-6, 6-0, 6-4 in two hours 14 minutes to give Germany an unassailable 3-0 lead over Australia.
The Australian pair of Casey Dellacqua and Ashleigh Barty prevented a wipe-out when they claimed the "dead" doubles rubber 6-2, 6-7 (5/7), 10-2 over Julia Goerges and Anna-Lena Grönefeld.
The final result was Germany 3, Australia 1. Germany will play either the Czech Republic or Italy in the final.
Left-hander Kerber showed great resilience to bounce back in the pivotal singles rubber and win seven games straight after the opening set to turn around the momentum, fighting off the 19th-ranked Stosur to take victory and put Germany into this year's final.
Germany last won the competition in 1992 when Steffi Graf and Anke Huber starred in the team which defeated Spain.
The Germans took a firm grip on the semi-final winning Saturday's opening two singles rubbers in the best-of-five match tie.
"It's a really special moment for us. We are all working hard to get to the final and win against Australia," Kerber said.
It was Kerber's first win in three encounters against Stosur.
"It was tough and I knew I had to play my best tennis to beat Sam because she is a Grand Slam champion," Kerber said.
She added; "After the first set I was just thinking about playing for the team and I was fighting until the last point. I was a little bit nervous on the last points but now I'm just happy. After 1992 we are now in the final and it means a lot for Germany and it's an honour for all of us."
Dellacqua's singles match with Andrea Petkovic was not played.