It turned out, that this problem was related to my website was running Umbraco CMS. Umbraco CMS has an AppSetting called "umbracoReservedPaths", which asks Umbraco to ignore specific paths.
The value was set to:
<add key="umbracoReservedPaths" value="~/umbraco,~/install/" />
After adding ~/api/, everything worked as expected:
<add key="umbracoReservedPaths" value="~/umbraco,~/install/,~/api/" />