Kerber overcame a slow start to win 4-6, 6-0, 6-4 and put Germany into its first Fed Cup final 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 hard courts at Pat Rafter Arena.
In the third game which went to eight deuces, Stosur's forehand error 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.
Germany will play either Czech Republic or Italy in the final on Nov. 8-9. Czech Republic led 2-0 after the first day.
On Saturday, Andrea Petkovic and Kerber had straight-set wins — Petkovic opened with a 6-1, 7-6 (7) win over Stosur while Kerber defeated Casey Dellacqua 6-1, 6-0 in 51 minutes.
In later matches Sunday, Dellacqua was scheduled to play Petkovic and Dellacqua and Ashleigh Barty were set to play Julia Goerges and Anna-Lena Groenefeld in doubles.
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.