Two objectives:

(1) When /page.html is entered, we want Nginx to redirect to /page. This can be done by:

location / {
  if ($request_uri ~ ^/(.*)\.html$) {
    return 302 /$1;
  }
}

(2) When a link /page is clicked, we want Nginx to serve static page named "page.html" in your www directory.

location / {
  try_files $uri $uri.html $uri/ =404;
}

(3) Combine both of these, you get the following location block:

location / {
  if ($request_uri ~ ^/(.*)\.html$) {
    return 302 /$1;
  }
  try_files $uri $uri.html $uri/ =404;
}

(4) Reload Nginx

sudo service nginx restart