Businesses use this language to allow for the possibility that your payment arrived before their letter or bill reached you. They’re telling you what their records indicate on the day they sent you the letter or bill.

If your payment should have been there well before the date given, contact the bank to see if your check has cleared.

EDIT: Good job, diesel, I like yours. Mine are always so late and a dollar short… I’ll get an invoice any day now!

The phrase is used to provide a “status report” relating to a current or future event.

In your example, “as of” means:

“The current situation, in the month of September, is that payment for the invoice has not been received.”

They received something in August. But they have not received payment since then.

This is a borderline gray-area phrase. Similar to those phrases that have a meaning that’s different from the actual words in the phrase.