This is now much easier in Apache 2.2.16+ using the FallbackResource directive.
FallbackResource /app/index.html
http://httpd.apache.org/docs/2.2/mod/mod_dir.html#fallbackresource
Depending on how you’re forwarding access to your API you may need to also leverage the enclosure to disable the fallback resource on specifically on API requests (2.2.24+).
<Directory /api>
FallbackResource disabled
</Directory>