You should use SQL Azure Database Import Export Service. You can find more about this service here.
For export functionallity you can then use this piece of code:
//Set Inputs to the REST Requests in the helper class
IEHelper.EndPointUri = @"<address of database endpoint according to its location, e.g. https://by1prod-dacsvc.azure.com/DACWebService.svc for DB located in West US>";
IEHelper.ServerName = "<database_server_name>.database.windows.net";
IEHelper.StorageKey = "<storage_key>";
IEHelper.DatabaseName = "<database_name>";
IEHelper.UserName = "<database_user_name>";
IEHelper.Password = "<database_password>";
//Do Export operation
string exportBlobPath = IEHelper.DoExport(String.Format(@"https://<storage_name>.blob.core.windows.net/bacpac/{0}.bacpac", IEHelper.DatabaseName));
ImportExportHelper IEHelper = new ImportExportHelper();