Microsoft Azure IaaS (Infrastructure as a Service) is picky when it comes to change DNS and network adapter settings at the VM level.
If for some reason you lost connectivity after playing with DNS/IP settings at the adapter level of your VM, then a good approach to solve the issue is redeploying the VM, this process will move the VM to a different host and by doing that the NIC interface will be reset.
The process is simple, inside of the VM properties, click on Redeploy under Support + Troubleshooting section.
In the new blade, confirm the redeploy process by hitting the Redeploy button. Keep in mind that the VM will be restarted as part of the process and it will take sometime for the VM to come back.
Wait the process to finish and then try to connect again.
If you want more information about DNS resolution in Microsoft Azure, this document is great start and it contains all scenarios and possible solutions, as follows: https://azure.microsoft.com/en-us/documentation/articles/virtual-networks-name-resolution-for-vms-and-role-instances/