It's a lot like what @smhinsey described. Team City was our CI platform (could be TFS), applied XML transformations using PowerShell for the target environment

It's a lot like what @smhinsey described. Team City was our CI platform (could be TFS), applied XML transformations using PowerShell for the target environment (I'd probably use the .config patching nowadays), built it using msbuild, which output a cloud deploy package, and using the Azure cmdlets deployed that package to the Staging slot, started it, waited for it to respond to a HTTP request, and then performed a VIP swap. We did have to take an outage for breaking schema changes. Publish to Staging slot stopped, stop the Prod slot, upgrade the database, start Staging, swap the VIP. Total outage was as long as it took to start the VM, a minute or two. We used code-first database with NHibernate.