Kerber overcame a slow start to win 4-6, 6-0, 6-4, giving Germany an unbeatable 3-0 lead and its first Fed Cup final berth since 1992, when it beat Spain for the title.
Stosur opened the third set by double-faulting on break point, but broke back to level on the hard court at Pat Rafter Arena.
In the third game, which went to eight deuces, Stosur's forehand that landed long gave Kerber the break and a 2-1 lead. Kerber was broken while serving for the match, but held serve two games later to close it out, clinching the semifinal with an ace.
"It's a very special moment for me and the whole team," Kerber said. "It feels very great. I played a good match but it was a tough one because Sam was playing also very tough and I have never beaten her before (in two previous matches)."
Germany will play either the Czech Republic or Italy in the final on Nov. 8-9. The Czechs led 2-0 after the first day.
A later reverse singles match between Casey Dellacqua and Andrea Petkovic was not played.
Dellacqua and Ashleigh Barty, who advanced to three Grand Slam doubles finals last year, beat Julia Goerges and Anna-Lena Groenefeld 6-2, 6-7 (5) and 10-2 in the deciding pro set to make the final scoreline 3-1.
Stosur said she gave it her best shot against Kerber on Sunday.
"I don't know if I can fight any harder than I did in that match," the former U.S. Open champion said. "In the second set I have to say I lost my way and that didn't help. I knew I had to regroup and start again. I had chances today to win it but that's what it comes down to playing a top player. It's not a lost opportunity, the only opportunity lost was not making the final and that was always going to be tough."
On Saturday, Petkovic and Kerber had straight-set wins — Petkovic opened with a 6-1, 7-6 (7) win over Stosur while Kerber defeated Dellacqua 6-1, 6-0.
Two years ago, Australia upset Germany 3-2 in Stuttgart in a World Group encounter in which Stosur won both her singles matches.
It was Australia's first semifinal appearance since 1993. Australia has not won the Fed Cup since 1974 when seven-time Grand Slam singles champion Evonne Goolagong led Australia to victory.