Angerer, goalkeeper for the national team, was chosen ahead of last year's winner Abby Wambach from the USA, and the five-times winner Marta from Brazil.
Neid, the national team's trainer, won the award for the second time, after first claiming it in 2010.
"I am surprised and proud, and thankful - everything at the moment actually. One never achieves such success alone," said Angerer. "I am very grateful to have played in this fantastic team in 2013, which made it very easy for me to perform."
Her award citation mentioned the 35-year-old's outstanding performance in Sweden last summer, when the German team won the European championship. Angerer saved two penalties in the final against Norway.
Neid also mentioned her teammates when accepting her award. "Now I stand here alone although so many people were part of it. I am delighted with this award, but also see it as recognition for all of German women's football," she said.
Real Madrid and Portugal attacker Cristiano Ronaldo won the FIFA Ballon d'Or, ending Lionel Messi's four-year run as the world's best player.