You can combine INSERT and DELETE in single statement:
DELETE [DB_Production].[dbo].[tableDetail]
output deleted.col1, deleted.col2, etc into [DB_Archive].[dbo].[tableDetail]
where exists (select * from [DB_Archive].[dbo].[tableMaster] tblMaster where [DB_Production].[dbo].[tableDetail].QuoNo = tblMaster.QuoNo)