After searching the web and having some contact with Microsoft I found the necessary configuration steps.
These are the things you have to do:
- Link the two websites to your team project
- Branch your code
- Adjust the workspace mappings on the build definition to the staging site to map the branch
- Change the solution to build property on the build definition for your staging site to point to the solution in your branch
I wrote a blog post about it that explains it in more detail: Branches, Team Foundation Service and Azure Websites