I built an Azure Virtual network with one VM being a Domain Controller and few other VMs being SharePoint 2013 related Servers. When I logged in to any of my Virtual Machine Windows Servers running on this virtual network, I was unable to open web sites from internet. 

Missing Microsoft external DNS server was the root cause. while creating the virtual network i opted to go with my domain controller rather than the MS public DNS, hence the problem.

You can edit the Virtual Network, under the configure section - select external in the drop-down. An ip address 168.63.129.16 will automatically get set. Save the change and you are done.