gistfile1.txt

server {
        listen 81;

        root /var/sites/app/webroot/;

        server_name localhost;

        rewrite_log on;

        location ^~ /api/ {
                rewrite /api/(.*)$ /$1 last;
        }

        location / {
                index index.php index.html index.htm;

                try_files $uri $uri/ /index.php?$uri&$args;
        }

        location ~ \.php$ {
                fastcgi_pass unix:/var/run/php5-fpm.sock;
                fastcgi_index index.php;
                include fastcgi_params;
                fastcgi_param SCRIPT_FILENAME $document_root$fastcgi_script_name;
        }

        # deny access to .htaccess files, if Apache's document root
        # concurs with nginx's one
        #
        location ~ /\.ht {
                deny all;
        }
}

did you try:

rewrite "/api/(.*)$" /$1 last;

instead?