I found a blog post that presented a very easy solution. Find the file that Nginx is serving from the "ssl_certificate" directive, and run the following commands on it (where "ssl.crt" is your certificate)
curl http://www.startssl.com/certs/sub.class1.server.ca.pem >>ssl.crt
curl http://www.startssl.com/certs/ca.pem >>ssl.crt
Restart Nginx, and you're ready to go!