When you create an Express site from Microsoft Azure's gallery, it includes some default setup in a web.config. I have some code adding Expires headers for content in my /images directory. I commented a Rewrite rule in web.config so server.js can handle the static assets correctly.

web.config

<configuration>
  <system.webServer>
    <rewrite>
      <rules>
        <!-- First we consider whether the incoming URL matches a physical file in the /public folder -->
        <rule name="StaticContent">
          <!--<action type="Rewrite" url="public{REQUEST_URI}"/>--> <!--Comment this to let Node handle your static files -->
          <action type="Rewrite" url="server.js"/>
        </rule>

        <!-- All other URLs are mapped to the Node.js application entry point -->
        <rule name="DynamicContent">
          <conditions>
            <add input="{REQUEST_FILENAME}" matchType="IsFile" negate="True"/>
          </conditions>
          <action type="Rewrite" url="server.js"/>
        </rule>
      </rules>
    </rewrite>
  </system.webServer>
</configuration>