Laurin - you are correct - while Websites can utilise Hybrid Connections to connect back to services on-premises they aren't actually able to connect (and be restricted to) internal Azure services.
If you use a Web Role you will need to setup a Virtual Network with an appropriate private IP address range and then ensure you add your Web Role to this Virtual Network. This is done by editing the service configuration of your Cloud Service deployment in Visual Studio and making it similar to the below:
<?xml version="1.0" encoding="utf-8"?>
<ServiceConfiguration ...>
<Role name="WebRole1">
...
</Role>
<NetworkConfiguration>
<Dns>
<DnsServers>
<DnsServer name="YourDns" IPAddress="10.4.3.1" />
</DnsServers>
</Dns>
<VirtualNetworkSite name="YourVirtualNetwork" />
<AddressAssignments>
<InstanceAddress roleName="WebRole1">
<Subnets>
<Subnet name="FrontEndSubnet" />
</Subnets>
</InstanceAddress>
</AddressAssignments>
</NetworkConfiguration>
</ServiceConfiguration>