The Problem

Part1 - Fault tolerence

A common configuration of web servers is setting up multiple servers to serve the same content, with a load-balancing method redirecting / forwarding clients to a specific server.
For instance, let's say I'm using servers server and otherserver to serve the site cluster-name.
The cheapest (as in "not requiring any additional software/hardware") way of doing it is creating the following DNS records:

cluster-name IN CNAME server
cluster-name IN CNAME otherserver

Due to round-robin DNS, about 50% of the clients querying for cluster-name will get the record pointing to server first (and therefor access it), and the other ones will get otherserver first. As long as the server and otherserver are configured to reply to requests containing Host:"cluster-nane" with the right application, everything works fine.

Part2 - Monitoring

A popular weay to monitor web applications is hitting a specific page and making sure it comes out OK (for example, fetching /test and making sure it's empty).
When you have just one server, this is easy to test. For example, one can use:

SERVER='http://site-adress'
[ -z $(curl -s "$SERVER/test") ] || echo '$SERVER is damaged'

However, when using a configuration with mutliple servers, you can't use the cluster name, because you'll get a random server, and as a responsible admin you want to test all of them individually.
When the servers redirect any hostname to this site, this is still relatively easy to script, like:

SERVERS=( 'server' 'otherserver' )
for SERVER in "${SERVERS[@]}"; do
    [ -z $(curl -s "http://$SERVER/test") ] || echo '$SERVER is damaged'
done

However, most sites are configured on virtual hosting, so they respond only to a specific hostname (mostly to allow serving several sites on the same port). When using the above script on such sites, it won't work (because the host name is wrong).

The Solution

Any proper solution should modify the client (the system running the monitoring script) and not the server, so I'm not talking about serving the site at another IP etc.

The primitive one

The most primitive way of accessing a specific server is by modifying the HOSTS file, adding a record like

<server ip>    cluster-name

However, modifying the HOSTS file requires root/admin privilleges, might affect other processes on the machine and feels messy.

The universal one

This method works with every tool I saw, because it's utilizing HTTP/Web Proxy support, which is pretty basic.
Normally, the client opens a socket to the hostname/port combination in the URL, writes the HTTP request and reads the response.
When using an HTTP proxy, the client opens a socket to the proxy hostname/port combination and reads the response. The proxy is in charge of contacting the actual host via the Host HTTP header.
By specifying the server as the proxy, we can keep the request "addressed" to cluster-name (by using the HTTP header Host: "cluster-name"), but actually access the server we want.
And now for some examples, assuming server:port is the actual server/port listening, and http://cluster-name/site is the target site.

Using cURL / wget

Both applications can use the environment variable http_proxy, so the syntax is quite elegant.

http_proxy='http://server:port' curl 'http://cluster-name/site'
http_proxy='http://server:port' wget 'http://cluster-name/site'

Using PowerShell

$prox = new-object System.Net.WebProxy 'http://server:port'
$req = [System.Net.WebRequest]::Create('http://cluster-name/site')
$req.Proxy = $prox
$req.GetResponse()

Using Python

Using the requests module

import requests
proxies = {
    'http': 'http://server:port'
}

requests.get('http://cluster-name/site', proxies=proxies)

Using Ruby

require 'net/http'
Net::HTTP.new('cluster-name', nil, 'server', 'port').start {
    |http| http.get('cluster-name/site')
}