I ran into this problem today and it worked for me:
Set the following environment variables (application settings) on your Azure website:
SCM_KRE_Version 1.0.0-beta2
SCM_KRE_CLR CoreCLR
. SCM_KRE_CLR "" clr, SCM_KRE_CLR CoreCLR, CoreCLR, ASPNETCORE50 ( DNXCORE50 3)
