How do I make Nginx redirect all requests for files which do not exist to a single php file?

Going by your additional comments this sounds like it might be the most optimal way, though it's not a pretty config.

server {
    listen 80 default_server;

    access_log /path/to/site/dir/logs/access.log;
    error_log /path/to/site/dir/logs/error.log;

    root /path/to/site/dir/webroot;
    index index.php index.html;

    location / {
        try_files $uri $uri/ /index.php;
    }

    location ~ \.php$ {
        try_files $uri @missing;

        fastcgi_pass localhost:9000;
        fastcgi_param SCRIPT_FILENAME $document_root$fastcgi_script_name;
        include /path/to/nginx/conf/fastcgi_params;
    }

    location @missing {
        rewrite ^ /error/404 break;

        fastcgi_pass localhost:9000;
        fastcgi_param SCRIPT_FILENAME $document_root/index.php;
        include /path/to/nginx/conf/fastcgi_params;
    }
}

Wow, I think all you want to replace that code is:

error_page 404 /index.php

...if I read what you want correctly.

What you are trying to do is same as custom error page. You can use the error_page property of nginx to achieve this. Follow the link for more information

http://wiki.nginx.org/HttpCoreModule#error_page