In VS2013, choose 'Deploy' from the context menu of the Web App project which is referenced by the role (and not the 'Cloud Service' project referencing the role(s) ... doing that will publish the whole service and all of its roles non-incrementally).
There should be a 'publish profile' which refers to the website on the *.cloudapp.net platform. If there is, it was put there by Visual Studio ... probably when you first ticked the 'Enable Web Deploy for all web roles' check-box when first deploying the service.
However, it might not be there, or be wrong. Because, for example, Visual Studio is dumb, and noticing that you had already published the Web App as a 'Web Site' (before you decided to incorporate it as a 'Web Role' in a service), it decided to leave the existing entry for *.azurewebsites.net alone. Or is it because that project was only included in the role as an <VirtualApplication /> element in the service definition's XML, and Visual Studio just neglected to create it?
So you'll have to find some way to force Visual Studio to re-create the right profile for you, or you'll have to manually copy and edit the publishing profile XML files from a known good source (Properties/PublishProfiles directory under the project root), or just guess the settings from absent documentation.
Those settings will be something like:
<?xml version="1.0" encoding="utf-8"?>
<Project ToolsVersion="12.0" xmlns="http://schemas.microsoft.com/developer/msbuild/2003">
<PropertyGroup>
<WebPublishMethod>MSDeploy</WebPublishMethod>
<MSDeployServiceURL>https://MYAPP.cloudapp.net:8172/MsDeploy.axd</MSDeployServiceURL>
<DeployIisAppPath>INSTANCE</DeployIisAppPath>
<RemoteSitePhysicalPath />
<AllowUntrustedCertificate>True</AllowUntrustedCertificate>
<SkipExtraFilesOnServer>True</SkipExtraFilesOnServer>
<DeployAsIisApp>False</DeployAsIisApp>
<MSDeployPublishMethod>WMSVC</MSDeployPublishMethod>
<UserName>MYUSER</UserName>
</PropertyGroup>
</Project>
Where:
MYAPP - is the domain name of the deployed Azure service
INSTANCE - is the name of the instance you want to update (get it from Server Explorer ... it's the 'Name' property of the 'Instance' under your service ... or get it from the list of instances comprising your service from the Windows Azure management dashboard).
MYUSER - is the name of the user you set up when publishing initially, for the Remote Desktop Services